9 - Julia is fast.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Julia is fast\n",
    "\n",
    "Very often, benchmarks are used to compare languages.  These benchmarks can lead to long discussions, first as to exactly what is being benchmarked and secondly what explains the differences.  These simple questions can sometimes get more complicated than you at first might imagine.\n",
    "\n",
    "The purpose of this notebook is for you to see a simple benchmark for yourself.  One can read the notebook and see what happened on the author's Macbook Pro with a 4-core Intel Core I7, or run the notebook yourself.\n",
    "\n",
    "(This material began life as a wonderful lecture by Steven Johnson at MIT: https://github.com/stevengj/18S096/blob/master/lectures/lecture1/Boxes-and-registers.ipynb.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Outline of this notebook\n",
    "\n",
    "- Define the sum function\n",
    "- Implementations & benchmarking of sum in...\n",
    "    - C (hand-written)\n",
    "    - C (hand-written with -ffast-math)\n",
    "    - python (built-in)\n",
    "    - python (numpy)\n",
    "    - python (hand-written)\n",
    "    - Julia (built-in)\n",
    "    - Julia (hand-written)\n",
    "    - Julia (hand-written with SIMD)\n",
    "- Summary of benchmarks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# `sum`: An easy enough function to understand"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the  **sum** function `sum(a)`, which computes\n",
    "$$\n",
    "\\mathrm{sum}(a) = \\sum_{i=1}^n a_i,\n",
    "$$\n",
    "where $n$ is the length of `a`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10000000-element Vector{Float64}:\n",
       " 0.5012541803539059\n",
       " 0.9933609203401452\n",
       " 0.3944893180676192\n",
       " 0.2446035212262776\n",
       " 0.8143940529013705\n",
       " 0.032887372314962526\n",
       " 0.8262733282856822\n",
       " 0.3912114710559349\n",
       " 0.43529567366601785\n",
       " 0.1542058799204118\n",
       " 0.05746977526966601\n",
       " 0.18317127845215508\n",
       " 0.6431322872083249\n",
       " ⋮\n",
       " 0.4063528564437253\n",
       " 0.026866494828949206\n",
       " 0.9339921977019192\n",
       " 0.310028849252449\n",
       " 0.739042234752626\n",
       " 0.1699746236340145\n",
       " 0.9504883470305974\n",
       " 0.7248007274709465\n",
       " 0.36192938797492036\n",
       " 0.7400797699007593\n",
       " 0.6721029424239575\n",
       " 0.06318143950068444"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = rand(10^7) # 1D vector of random numbers, uniform on [0,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.001596529905331e6"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expected result is 0.5 * 10^7, since the mean of each entry is 0.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Benchmarking a few ways in a few languages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.005433 seconds (1 allocation: 16 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "5.001596529905331e6"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.006755 seconds (1 allocation: 16 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "5.001596529905331e6"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.005820 seconds (1 allocation: 16 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "5.001596529905331e6"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time sum(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `@time` macro can yield noisy results, so it's not our best choice for benchmarking!\n",
    "\n",
    "Luckily, Julia has a `BenchmarkTools.jl` package to make benchmarking easy and accurate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Pkg\n",
    "Pkg.add(\"BenchmarkTools\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "using BenchmarkTools  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#  1. The C language\n",
    "\n",
    "C is often considered the gold standard: difficult on the human, nice for the machine. Getting within a factor of 2 of C is often satisfying. Nonetheless, even within C, there are many kinds of optimizations possible that a naive C writer may or may not get the advantage of.\n",
    "\n",
    "The current author does not speak C, so he does not read the cell below, but is happy to know that you can put C code in a Julia session, compile it, and run it. Note that the `\"\"\"` wrap a multi-line string."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "c_sum (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Libdl\n",
    "C_code = \"\"\"\n",
    "#include <stddef.h>\n",
    "double c_sum(size_t n, double *X) {\n",
    "    double s = 0.0;\n",
    "    for (size_t i = 0; i < n; ++i) {\n",
    "        s += X[i];\n",
    "    }\n",
    "    return s;\n",
    "}\n",
    "\"\"\"\n",
    "\n",
    "const Clib = tempname()   # make a temporary file\n",
    "\n",
    "\n",
    "# compile to a shared library by piping C_code to gcc\n",
    "# (works only if you have gcc installed):\n",
    "\n",
    "open(`gcc -fPIC -O3 -msse3 -xc -shared -o $(Clib * \".\" * Libdl.dlext) -`, \"w\") do f\n",
    "    print(f, C_code) \n",
    "end\n",
    "\n",
    "# define a Julia function that calls the C function:\n",
    "c_sum(X::Array{Float64}) = ccall((\"c_sum\", Clib), Float64, (Csize_t, Ptr{Float64}), length(X), X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.001596529905427e6"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_sum(a) ≈ sum(a) # type \\approx and then <TAB> to get the ≈ symbolb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.592622518539429e-8"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_sum(a) - sum(a)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "isapprox (generic function with 9 methods)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "≈  # alias for the `isapprox` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "search: \u001b[0m\u001b[1mi\u001b[22m\u001b[0m\u001b[1ms\u001b[22m\u001b[0m\u001b[1ma\u001b[22m\u001b[0m\u001b[1mp\u001b[22m\u001b[0m\u001b[1mp\u001b[22m\u001b[0m\u001b[1mr\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1mx\u001b[22m\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "\\begin{verbatim}\n",
       "isapprox(x, y; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps, nans::Bool=false[, norm::Function])\n",
       "\\end{verbatim}\n",
       "Inexact equality comparison: \\texttt{true} if \\texttt{norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))}. The default \\texttt{atol} is zero and the default \\texttt{rtol} depends on the types of \\texttt{x} and \\texttt{y}. The keyword argument \\texttt{nans} determines whether or not NaN values are considered equal (defaults to false).\n",
       "\n",
       "For real or complex floating-point values, if an \\texttt{atol > 0} is not specified, \\texttt{rtol} defaults to the square root of \\href{@ref}{\\texttt{eps}} of the type of \\texttt{x} or \\texttt{y}, whichever is bigger (least precise). This corresponds to requiring equality of about half of the significand digits. Otherwise, e.g. for integer arguments or if an \\texttt{atol > 0} is supplied, \\texttt{rtol} defaults to zero.\n",
       "\n",
       "The \\texttt{norm} keyword defaults to \\texttt{abs} for numeric \\texttt{(x,y)} and to \\texttt{LinearAlgebra.norm} for arrays (where an alternative \\texttt{norm} choice is sometimes useful). When \\texttt{x} and \\texttt{y} are arrays, if \\texttt{norm(x-y)} is not finite (i.e. \\texttt{±Inf} or \\texttt{NaN}), the comparison falls back to checking whether all elements of \\texttt{x} and \\texttt{y} are approximately equal component-wise.\n",
       "\n",
       "The binary operator \\texttt{≈} is equivalent to \\texttt{isapprox} with the default arguments, and \\texttt{x ≉ y} is equivalent to \\texttt{!isapprox(x,y)}.\n",
       "\n",
       "Note that \\texttt{x ≈ 0} (i.e., comparing to zero with the default tolerances) is equivalent to \\texttt{x == 0} since the default \\texttt{atol} is \\texttt{0}.  In such cases, you should either supply an appropriate \\texttt{atol} (or use \\texttt{norm(x) ≤ atol}) or rearrange your code (e.g. use \\texttt{x ≈ y} rather than \\texttt{x - y ≈ 0}).   It is not possible to pick a nonzero \\texttt{atol} automatically because it depends on the overall scaling (the \"units\") of your problem: for example, in \\texttt{x - y ≈ 0}, \\texttt{atol=1e-9} is an absurdly small tolerance if \\texttt{x} is the \\href{https://en.wikipedia.org/wiki/Earth_radius}{radius of the Earth} in meters, but an absurdly large tolerance if \\texttt{x} is the \\href{https://en.wikipedia.org/wiki/Bohr_radius}{radius of a Hydrogen atom} in meters.\n",
       "\n",
       "\\begin{quote}\n",
       "\\textbf{compat}\n",
       "\n",
       "Julia 1.6\n",
       "\n",
       "Passing the \\texttt{norm} keyword argument when comparing numeric (non-array) arguments requires Julia 1.6 or later.\n",
       "\n",
       "\\end{quote}\n",
       "\\section{Examples}\n",
       "\\begin{verbatim}\n",
       "julia> 0.1 ≈ (0.1 - 1e-10)\n",
       "true\n",
       "\n",
       "julia> isapprox(10, 11; atol = 2)\n",
       "true\n",
       "\n",
       "julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0])\n",
       "true\n",
       "\n",
       "julia> 1e-10 ≈ 0\n",
       "false\n",
       "\n",
       "julia> isapprox(1e-10, 0, atol=1e-8)\n",
       "true\n",
       "\\end{verbatim}\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\begin{verbatim}\n",
       "isapprox(x; kwargs...) / ≈(x; kwargs...)\n",
       "\\end{verbatim}\n",
       "Create a function that compares its argument to \\texttt{x} using \\texttt{≈}, i.e. a function equivalent to \\texttt{y -> y ≈ x}.\n",
       "\n",
       "The keyword arguments supported here are the same as those in the 2-argument \\texttt{isapprox}.\n",
       "\n"
      ],
      "text/markdown": [
       "```\n",
       "isapprox(x, y; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps, nans::Bool=false[, norm::Function])\n",
       "```\n",
       "\n",
       "Inexact equality comparison: `true` if `norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))`. The default `atol` is zero and the default `rtol` depends on the types of `x` and `y`. The keyword argument `nans` determines whether or not NaN values are considered equal (defaults to false).\n",
       "\n",
       "For real or complex floating-point values, if an `atol > 0` is not specified, `rtol` defaults to the square root of [`eps`](@ref) of the type of `x` or `y`, whichever is bigger (least precise). This corresponds to requiring equality of about half of the significand digits. Otherwise, e.g. for integer arguments or if an `atol > 0` is supplied, `rtol` defaults to zero.\n",
       "\n",
       "The `norm` keyword defaults to `abs` for numeric `(x,y)` and to `LinearAlgebra.norm` for arrays (where an alternative `norm` choice is sometimes useful). When `x` and `y` are arrays, if `norm(x-y)` is not finite (i.e. `±Inf` or `NaN`), the comparison falls back to checking whether all elements of `x` and `y` are approximately equal component-wise.\n",
       "\n",
       "The binary operator `≈` is equivalent to `isapprox` with the default arguments, and `x ≉ y` is equivalent to `!isapprox(x,y)`.\n",
       "\n",
       "Note that `x ≈ 0` (i.e., comparing to zero with the default tolerances) is equivalent to `x == 0` since the default `atol` is `0`.  In such cases, you should either supply an appropriate `atol` (or use `norm(x) ≤ atol`) or rearrange your code (e.g. use `x ≈ y` rather than `x - y ≈ 0`).   It is not possible to pick a nonzero `atol` automatically because it depends on the overall scaling (the \"units\") of your problem: for example, in `x - y ≈ 0`, `atol=1e-9` is an absurdly small tolerance if `x` is the [radius of the Earth](https://en.wikipedia.org/wiki/Earth_radius) in meters, but an absurdly large tolerance if `x` is the [radius of a Hydrogen atom](https://en.wikipedia.org/wiki/Bohr_radius) in meters.\n",
       "\n",
       "!!! compat \"Julia 1.6\"\n",
       "    Passing the `norm` keyword argument when comparing numeric (non-array) arguments requires Julia 1.6 or later.\n",
       "\n",
       "\n",
       "# Examples\n",
       "\n",
       "```jldoctest\n",
       "julia> 0.1 ≈ (0.1 - 1e-10)\n",
       "true\n",
       "\n",
       "julia> isapprox(10, 11; atol = 2)\n",
       "true\n",
       "\n",
       "julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0])\n",
       "true\n",
       "\n",
       "julia> 1e-10 ≈ 0\n",
       "false\n",
       "\n",
       "julia> isapprox(1e-10, 0, atol=1e-8)\n",
       "true\n",
       "```\n",
       "\n",
       "---\n",
       "\n",
       "```\n",
       "isapprox(x; kwargs...) / ≈(x; kwargs...)\n",
       "```\n",
       "\n",
       "Create a function that compares its argument to `x` using `≈`, i.e. a function equivalent to `y -> y ≈ x`.\n",
       "\n",
       "The keyword arguments supported here are the same as those in the 2-argument `isapprox`.\n"
      ],
      "text/plain": [
       "\u001b[36m  isapprox(x, y; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps, nans::Bool=false[, norm::Function])\u001b[39m\n",
       "\n",
       "  Inexact equality comparison: \u001b[36mtrue\u001b[39m if \u001b[36mnorm(x-y) <= max(atol,\n",
       "  rtol*max(norm(x), norm(y)))\u001b[39m. The default \u001b[36matol\u001b[39m is zero and the default \u001b[36mrtol\u001b[39m\n",
       "  depends on the types of \u001b[36mx\u001b[39m and \u001b[36my\u001b[39m. The keyword argument \u001b[36mnans\u001b[39m determines\n",
       "  whether or not NaN values are considered equal (defaults to false).\n",
       "\n",
       "  For real or complex floating-point values, if an \u001b[36matol > 0\u001b[39m is not specified,\n",
       "  \u001b[36mrtol\u001b[39m defaults to the square root of \u001b[36meps\u001b[39m of the type of \u001b[36mx\u001b[39m or \u001b[36my\u001b[39m, whichever is\n",
       "  bigger (least precise). This corresponds to requiring equality of about half\n",
       "  of the significand digits. Otherwise, e.g. for integer arguments or if an\n",
       "  \u001b[36matol > 0\u001b[39m is supplied, \u001b[36mrtol\u001b[39m defaults to zero.\n",
       "\n",
       "  The \u001b[36mnorm\u001b[39m keyword defaults to \u001b[36mabs\u001b[39m for numeric \u001b[36m(x,y)\u001b[39m and to \u001b[36mLinearAlgebra.norm\u001b[39m\n",
       "  for arrays (where an alternative \u001b[36mnorm\u001b[39m choice is sometimes useful). When \u001b[36mx\u001b[39m\n",
       "  and \u001b[36my\u001b[39m are arrays, if \u001b[36mnorm(x-y)\u001b[39m is not finite (i.e. \u001b[36m±Inf\u001b[39m or \u001b[36mNaN\u001b[39m), the\n",
       "  comparison falls back to checking whether all elements of \u001b[36mx\u001b[39m and \u001b[36my\u001b[39m are\n",
       "  approximately equal component-wise.\n",
       "\n",
       "  The binary operator \u001b[36m≈\u001b[39m is equivalent to \u001b[36misapprox\u001b[39m with the default arguments,\n",
       "  and \u001b[36mx ≉ y\u001b[39m is equivalent to \u001b[36m!isapprox(x,y)\u001b[39m.\n",
       "\n",
       "  Note that \u001b[36mx ≈ 0\u001b[39m (i.e., comparing to zero with the default tolerances) is\n",
       "  equivalent to \u001b[36mx == 0\u001b[39m since the default \u001b[36matol\u001b[39m is \u001b[36m0\u001b[39m. In such cases, you should\n",
       "  either supply an appropriate \u001b[36matol\u001b[39m (or use \u001b[36mnorm(x) ≤ atol\u001b[39m) or rearrange your\n",
       "  code (e.g. use \u001b[36mx ≈ y\u001b[39m rather than \u001b[36mx - y ≈ 0\u001b[39m). It is not possible to pick a\n",
       "  nonzero \u001b[36matol\u001b[39m automatically because it depends on the overall scaling (the\n",
       "  \"units\") of your problem: for example, in \u001b[36mx - y ≈ 0\u001b[39m, \u001b[36matol=1e-9\u001b[39m is an\n",
       "  absurdly small tolerance if \u001b[36mx\u001b[39m is the radius of the Earth\n",
       "  (https://en.wikipedia.org/wiki/Earth_radius) in meters, but an absurdly\n",
       "  large tolerance if \u001b[36mx\u001b[39m is the radius of a Hydrogen atom\n",
       "  (https://en.wikipedia.org/wiki/Bohr_radius) in meters.\n",
       "\n",
       "\u001b[39m\u001b[1m  │ \u001b[22m\u001b[39m\u001b[1mJulia 1.6\u001b[22m\n",
       "\u001b[39m\u001b[1m  │\u001b[22m\n",
       "\u001b[39m\u001b[1m  │\u001b[22m  Passing the \u001b[36mnorm\u001b[39m keyword argument when comparing numeric\n",
       "\u001b[39m\u001b[1m  │\u001b[22m  (non-array) arguments requires Julia 1.6 or later.\n",
       "\n",
       "\u001b[1m  Examples\u001b[22m\n",
       "\u001b[1m  ≡≡≡≡≡≡≡≡≡≡\u001b[22m\n",
       "\n",
       "\u001b[36m  julia> 0.1 ≈ (0.1 - 1e-10)\u001b[39m\n",
       "\u001b[36m  true\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> isapprox(10, 11; atol = 2)\u001b[39m\n",
       "\u001b[36m  true\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> isapprox([10.0^9, 1.0], [10.0^9, 2.0])\u001b[39m\n",
       "\u001b[36m  true\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> 1e-10 ≈ 0\u001b[39m\n",
       "\u001b[36m  false\u001b[39m\n",
       "\u001b[36m  \u001b[39m\n",
       "\u001b[36m  julia> isapprox(1e-10, 0, atol=1e-8)\u001b[39m\n",
       "\u001b[36m  true\u001b[39m\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\u001b[36m  isapprox(x; kwargs...) / ≈(x; kwargs...)\u001b[39m\n",
       "\n",
       "  Create a function that compares its argument to \u001b[36mx\u001b[39m using \u001b[36m≈\u001b[39m, i.e. a function\n",
       "  equivalent to \u001b[36my -> y ≈ x\u001b[39m.\n",
       "\n",
       "  The keyword arguments supported here are the same as those in the 2-argument\n",
       "  \u001b[36misapprox\u001b[39m."
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?isapprox"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now benchmark the C code directly from Julia:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "c_bench = @benchmark c_sum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "C: Fastest time was 11.610195 msec\n"
     ]
    }
   ],
   "source": [
    "println(\"C: Fastest time was $(minimum(c_bench.times) / 1e6) msec\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 1 entry:\n",
       "  \"C\" => 11.6102"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = Dict()  # a \"dictionary\", i.e. an associative array\n",
    "d[\"C\"] = minimum(c_bench.times) / 1e6  # in milliseconds\n",
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "gr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
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       "</svg>\n"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Statistics # bring in statistical support for standard deviations\n",
    "t = c_bench.times / 1e6 # times in milliseconds\n",
    "m, σ = minimum(t), std(t)\n",
    "\n",
    "histogram(t, bins=500,\n",
    "    xlim=(m - 0.01, m + σ),\n",
    "    xlabel=\"milliseconds\", ylabel=\"count\", label=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. C with -ffast-math\n",
    "\n",
    "If we allow C to re-arrange the floating point operations, then it'll vectorize with SIMD (single instruction, multiple data) instructions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "c_sum_fastmath (generic function with 1 method)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "const Clib_fastmath = tempname()   # make a temporary file\n",
    "\n",
    "# The same as above but with a -ffast-math flag added\n",
    "open(`gcc -fPIC -O3 -msse3 -xc -shared -ffast-math -o $(Clib_fastmath * \".\" * Libdl.dlext) -`, \"w\") do f\n",
    "    print(f, C_code) \n",
    "end\n",
    "\n",
    "# define a Julia function that calls the C function:\n",
    "c_sum_fastmath(X::Array{Float64}) = ccall((\"c_sum\", Clib_fastmath), Float64, (Csize_t, Ptr{Float64}), length(X), X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 851 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m4.377 ms\u001b[22m\u001b[39m … \u001b[35m31.716 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m5.033 ms              \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m5.852 ms\u001b[22m\u001b[39m ± \u001b[32m 1.969 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[34m█\u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[34m█\u001b[39m\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▁\n",
       "  5.11 ms\u001b[90m        Histogram: frequency by time\u001b[39m        4.65 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m0 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m0\u001b[39m."
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_fastmath_bench = @benchmark $c_sum_fastmath($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.377002"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"C -ffast-math\"] = minimum(c_fastmath_bench.times) / 1e6  # in milliseconds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3. Python's built in `sum` "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `PyCall` package provides a Julia interface to Python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "using Pkg; Pkg.add(\"PyCall\")\n",
    "using PyCall"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PyObject <built-in function sum>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# get the Python built-in \"sum\" function:\n",
    "pysum = pybuiltin(\"sum\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.001596529905427e6"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pysum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pysum(a) ≈ sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 7 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m724.151 ms\u001b[22m\u001b[39m … \u001b[35m900.215 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m808.521 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m801.504 ms\u001b[22m\u001b[39m ± \u001b[32m 75.186 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[39m█\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m█\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[39m█\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m█\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▁\n",
       "  724 ms\u001b[90m           Histogram: frequency by time\u001b[39m          731 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m336 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m6\u001b[39m."
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "py_list_bench = @benchmark $pysum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 3 entries:\n",
       "  \"C\"               => 11.6102\n",
       "  \"Python built-in\" => 724.151\n",
       "  \"C -ffast-math\"   => 4.377"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Python built-in\"] = minimum(py_list_bench.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4. Python: `numpy` \n",
    "\n",
    "## Takes advantage of hardware \"SIMD\", but only works when it works.\n",
    "\n",
    "`numpy` is an optimized C library, callable from Python.\n",
    "It may be installed within Julia as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Pkg; Pkg.add(\"Conda\")\n",
    "using Conda"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Conda.add(\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 1017 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m4.204 ms\u001b[22m\u001b[39m … \u001b[35m32.455 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m4.622 ms              \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m4.903 ms\u001b[22m\u001b[39m ± \u001b[32m 1.281 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[34m█\u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[34m█\u001b[39m\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▁\n",
       "  5.35 ms\u001b[90m        Histogram: frequency by time\u001b[39m        4.55 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m336 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m6\u001b[39m."
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numpy_sum = pyimport(\"numpy\")[\"sum\"]\n",
    "\n",
    "py_numpy_bench = @benchmark $numpy_sum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.001596529905329e6"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numpy_sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numpy_sum(a) ≈ sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 4 entries:\n",
       "  \"C\"               => 11.6102\n",
       "  \"Python numpy\"    => 4.20387\n",
       "  \"Python built-in\" => 724.151\n",
       "  \"C -ffast-math\"   => 4.377"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Python numpy\"] = minimum(py_numpy_bench.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5. Python, hand-written "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PyObject <function py_sum at 0x7fd4872084c0>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "py\"\"\"\n",
    "def py_sum(A):\n",
    "    s = 0.0\n",
    "    for a in A:\n",
    "        s += a\n",
    "    return s\n",
    "\"\"\"\n",
    "\n",
    "sum_py = py\"py_sum\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 6 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m941.357 ms\u001b[22m\u001b[39m … \u001b[35m995.683 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m969.960 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m968.470 ms\u001b[22m\u001b[39m ± \u001b[32m 23.532 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[34m \u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[34m▁\u001b[39m\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▁\n",
       "  992 ms\u001b[90m           Histogram: frequency by time\u001b[39m          965 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m336 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m6\u001b[39m."
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "py_hand = @benchmark $sum_py($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.001596529905427e6"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum_py(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum_py(a) ≈ sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 5 entries:\n",
       "  \"C\"                   => 11.6102\n",
       "  \"Python numpy\"        => 4.20387\n",
       "  \"Python hand-written\" => 941.357\n",
       "  \"Python built-in\"     => 724.151\n",
       "  \"C -ffast-math\"       => 4.377"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Python hand-written\"] = minimum(py_hand.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 6. Julia (built-in) \n",
    "\n",
    "## Written directly in Julia, not in C!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "sum(a::<b>AbstractArray</b>; <i>dims, kw...</i>) in Base at <a href=\"https://github.com/JuliaLang/julia/tree/1b93d53fc4bb59350ada898038ed4de2994cce33/base/reducedim.jl#L873\" target=\"_blank\">reducedim.jl:873</a>"
      ],
      "text/plain": [
       "sum(a::AbstractArray; dims, kw...) in Base at reducedim.jl:873"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@which sum(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 1048 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m4.170 ms\u001b[22m\u001b[39m … \u001b[35m  9.967 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m4.419 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m4.752 ms\u001b[22m\u001b[39m ± \u001b[32m900.697 μs\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[34m█\u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[34m█\u001b[39m\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▁\n",
       "  5 ms\u001b[90m            Histogram: frequency by time\u001b[39m         4.4 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m0 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m0\u001b[39m."
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j_bench = @benchmark sum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 6 entries:\n",
       "  \"C\"                   => 11.6102\n",
       "  \"Python numpy\"        => 4.20387\n",
       "  \"Python hand-written\" => 941.357\n",
       "  \"Python built-in\"     => 724.151\n",
       "  \"Julia built-in\"      => 4.17017\n",
       "  \"C -ffast-math\"       => 4.377"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Julia built-in\"] = minimum(j_bench.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7. Julia (hand-written) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mysum (generic function with 1 method)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mysum(A)   \n",
    "    s = 0.0 # s = zero(eltype(a))\n",
    "    for a in A\n",
    "        s += a\n",
    "    end\n",
    "    s\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "j_bench_hand = @benchmark mysum($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 7 entries:\n",
       "  \"C\"                   => 11.6102\n",
       "  \"Python numpy\"        => 4.20387\n",
       "  \"Julia hand-written\"  => 11.8939\n",
       "  \"Python hand-written\" => 941.357\n",
       "  \"Python built-in\"     => 724.151\n",
       "  \"Julia built-in\"      => 4.17017\n",
       "  \"C -ffast-math\"       => 4.377"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Julia hand-written\"] = minimum(j_bench_hand.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 8. Julia (hand-written w. simd) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mysum_simd (generic function with 1 method)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mysum_simd(A)   \n",
    "    s = 0.0 # s = zero(eltype(A))\n",
    "    @simd for a in A\n",
    "        @inbounds s += a\n",
    "    end\n",
    "    s\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: 1047 samples with 1 evaluation.\n",
       " Range \u001b[90m(\u001b[39m\u001b[36m\u001b[1mmin\u001b[22m\u001b[39m … \u001b[35mmax\u001b[39m\u001b[90m):  \u001b[39m\u001b[36m\u001b[1m4.086 ms\u001b[22m\u001b[39m … \u001b[35m  9.585 ms\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmin … max\u001b[90m): \u001b[39m0.00% … 0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[34m\u001b[1mmedian\u001b[22m\u001b[39m\u001b[90m):     \u001b[39m\u001b[34m\u001b[1m4.594 ms               \u001b[22m\u001b[39m\u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmedian\u001b[90m):    \u001b[39m0.00%\n",
       " Time  \u001b[90m(\u001b[39m\u001b[32m\u001b[1mmean\u001b[22m\u001b[39m ± \u001b[32mσ\u001b[39m\u001b[90m):   \u001b[39m\u001b[32m\u001b[1m4.757 ms\u001b[22m\u001b[39m ± \u001b[32m613.184 μs\u001b[39m  \u001b[90m┊\u001b[39m GC \u001b[90m(\u001b[39mmean ± σ\u001b[90m):  \u001b[39m0.00% ± 0.00%\n",
       "\n",
       "  \u001b[34m█\u001b[39m\u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \u001b[39m \n",
       "  \u001b[34m█\u001b[39m\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m▁\u001b[39m \u001b[39m▂\n",
       "  4.96 ms\u001b[90m         Histogram: frequency by time\u001b[39m        4.93 ms \u001b[0m\u001b[1m<\u001b[22m\n",
       "\n",
       " Memory estimate\u001b[90m: \u001b[39m\u001b[33m0 bytes\u001b[39m, allocs estimate\u001b[90m: \u001b[39m\u001b[33m0\u001b[39m."
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j_bench_hand_simd = @benchmark mysum_simd($a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.0015965299053285e6"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mysum_simd(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 8 entries:\n",
       "  \"Julia hand-written simd\" => 4.08637\n",
       "  \"C\"                       => 11.6102\n",
       "  \"Python numpy\"            => 4.20387\n",
       "  \"Julia hand-written\"      => 11.8939\n",
       "  \"Python hand-written\"     => 941.357\n",
       "  \"Python built-in\"         => 724.151\n",
       "  \"Julia built-in\"          => 4.17017\n",
       "  \"C -ffast-math\"           => 4.377"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d[\"Julia hand-written simd\"] = minimum(j_bench_hand_simd.times) / 1e6\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Summary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Julia hand-written simd.......4.076\n",
      "Julia built-in...............4.1702\n",
      "Python numpy.................4.2039\n",
      "C -ffast-math.................4.377\n",
      "C...........................11.6102\n",
      "Julia hand-written..........11.8939\n",
      "Python built-in............724.1514\n",
      "Python hand-written........941.3575\n"
     ]
    }
   ],
   "source": [
    "for (key, value) in sort(collect(d), by=last)\n",
    "    println(rpad(key, 25, \".\"), lpad(round(value; digits=4), 10, \".\"))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "α = 3\n",
    "∇f(u) = α*u; ∇f(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[90m;  @ int.jl:88 within `*'\u001b[39m\n",
      "\u001b[95mdefine\u001b[39m \u001b[36mi64\u001b[39m \u001b[93m@\"julia_\u001b[39m\u001b[0m*\u001b[0m_7137\"\u001b[33m(\u001b[39m\u001b[36mi64\u001b[39m \u001b[95msignext\u001b[39m \u001b[0m%0\u001b[0m, \u001b[36mi64\u001b[39m \u001b[95msignext\u001b[39m \u001b[0m%1\u001b[33m)\u001b[39m \u001b[33m{\u001b[39m\n",
      "\u001b[91mtop:\u001b[39m\n",
      "  \u001b[0m%2 \u001b[0m= \u001b[96m\u001b[1mmul\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%1\u001b[0m, \u001b[0m%0\n",
      "  \u001b[96m\u001b[1mret\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%2\n",
      "\u001b[33m}\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "@code_llvm 2*5\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.03125"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2^-5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "32"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2^5\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.03125"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2^-5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "^(x::<b>T</b>, p::<b>T</b>)<i> where T<:Integer</i> in Base at <a href=\"https://github.com/JuliaLang/julia/tree/1b93d53fc4bb59350ada898038ed4de2994cce33/base/intfuncs.jl#L290\" target=\"_blank\">intfuncs.jl:290</a>"
      ],
      "text/plain": [
       "^(x::T, p::T) where T<:Integer in Base at intfuncs.jl:290"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@which 2^-5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sumofsins1 (generic function with 1 method)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sumofsins1(n::Integer)\n",
    "    r = 0\n",
    "    for i in 1:n\n",
    "        r += sin(3.4)\n",
    "    end\n",
    "    return r\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10000"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "100_00"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-25554.110202663698"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sumofsins1(100_000)  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sumofsins2 (generic function with 1 method)"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sumofsins2(n::Integer)\n",
    "    r = 0.0\n",
    "    for i in 1:n\n",
    "        r += sin(3.4)\n",
    "    end\n",
    "    return r\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-25554.110202663698"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sumofsins2(100_000)  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.402020 seconds (114.72 k allocations: 7.148 MiB, 67.37% gc time, 93.23% compilation time)\n"
     ]
    }
   ],
   "source": [
    "@time [sumofsins1(100_000) for i in 1:100];  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.107550 seconds (110.03 k allocations: 6.921 MiB, 74.05% compilation time)\n"
     ]
    }
   ],
   "source": [
    "@time [sumofsins2(100_000) for i in 1:100];  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.077988 seconds (110.03 k allocations: 6.921 MiB, 83.72% compilation time)\n"
     ]
    }
   ],
   "source": [
    "@time [sumofsins2(100_000) for i in 1:100];  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.075641 seconds (110.03 k allocations: 6.921 MiB, 83.47% compilation time)\n"
     ]
    }
   ],
   "source": [
    "@time [sumofsins2(100_000) for i in 1:100];  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.075624 seconds (110.03 k allocations: 6.921 MiB, 83.07% compilation time)\n"
     ]
    }
   ],
   "source": [
    "@time [sumofsins2(100_000) for i in 1:100];  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.085725 seconds (114.69 k allocations: 7.146 MiB, 70.72% compilation time)\n"
     ]
    }
   ],
   "source": [
    "@time [sumofsins1(100_000) for i in 1:100];  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.084017 seconds (114.69 k allocations: 7.146 MiB, 70.01% compilation time)\n"
     ]
    }
   ],
   "source": [
    "@time [sumofsins1(100_000) for i in 1:100];  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[90m;  @ In[64]:1 within `sumofsins1'\u001b[39m\n",
      "\u001b[95mdefine\u001b[39m \u001b[33m{\u001b[39m \u001b[33m{\u001b[39m\u001b[33m}\u001b[39m\u001b[0m*\u001b[0m, \u001b[36mi8\u001b[39m \u001b[33m}\u001b[39m \u001b[93m@julia_sumofsins1_7314\u001b[39m\u001b[33m(\u001b[39m\u001b[33m[\u001b[39m\u001b[33m8\u001b[39m \u001b[0mx \u001b[36mi8\u001b[39m\u001b[33m]\u001b[39m\u001b[0m* \u001b[95mnoalias\u001b[39m \u001b[95mnocapture\u001b[39m \u001b[95malign\u001b[39m \u001b[33m8\u001b[39m \u001b[95mdereferenceable\u001b[39m\u001b[33m(\u001b[39m\u001b[33m8\u001b[39m\u001b[33m)\u001b[39m \u001b[0m%0\u001b[0m, \u001b[36mi64\u001b[39m \u001b[95msignext\u001b[39m \u001b[0m%1\u001b[33m)\u001b[39m \u001b[33m{\u001b[39m\n",
      "\u001b[91mtop:\u001b[39m\n",
      "\u001b[90m;  @ In[64]:3 within `sumofsins1'\u001b[39m\n",
      "\u001b[90m; ┌ @ range.jl:5 within `Colon'\u001b[39m\n",
      "\u001b[90m; │┌ @ range.jl:287 within `UnitRange'\u001b[39m\n",
      "\u001b[90m; ││┌ @ range.jl:292 within `unitrange_last'\u001b[39m\n",
      "     \u001b[0m%.inv \u001b[0m= \u001b[96m\u001b[1micmp\u001b[22m\u001b[39m \u001b[96m\u001b[1msgt\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%1\u001b[0m, \u001b[33m0\u001b[39m\n",
      "     \u001b[0m%2 \u001b[0m= \u001b[96m\u001b[1mselect\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[0m%.inv\u001b[0m, \u001b[36mi64\u001b[39m \u001b[0m%1\u001b[0m, \u001b[36mi64\u001b[39m \u001b[33m0\u001b[39m\n",
      "\u001b[90m; └└└\u001b[39m\n",
      "  \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[0m%.inv\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L11\u001b[39m\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%union_move8\u001b[39m\n",
      "\n",
      "\u001b[91mL11:\u001b[39m                                              \u001b[90m; preds = %L49, %top\u001b[39m\n",
      "\u001b[90m; ┌ @ range.jl:670 within `iterate'\u001b[39m\n",
      "   \u001b[0m%.sroa.014.0 \u001b[0m= \u001b[96m\u001b[1mphi\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[33m[\u001b[39m \u001b[0m%8\u001b[0m, \u001b[91m%L49\u001b[39m \u001b[33m]\u001b[39m\u001b[0m, \u001b[33m[\u001b[39m \u001b[33m0\u001b[39m\u001b[0m, \u001b[91m%top\u001b[39m \u001b[33m]\u001b[39m\n",
      "   \u001b[0m%value_phi2 \u001b[0m= \u001b[96m\u001b[1mphi\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[33m[\u001b[39m \u001b[0m%7\u001b[0m, \u001b[91m%L49\u001b[39m \u001b[33m]\u001b[39m\u001b[0m, \u001b[33m[\u001b[39m \u001b[33m1\u001b[39m\u001b[0m, \u001b[91m%top\u001b[39m \u001b[33m]\u001b[39m\n",
      "   \u001b[0m%3 \u001b[0m= \u001b[96m\u001b[1mphi\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[33m[\u001b[39m \u001b[95mfalse\u001b[39m\u001b[0m, \u001b[91m%L49\u001b[39m \u001b[33m]\u001b[39m\u001b[0m, \u001b[33m[\u001b[39m \u001b[95mtrue\u001b[39m\u001b[0m, \u001b[91m%top\u001b[39m \u001b[33m]\u001b[39m\n",
      "   \u001b[0m%4 \u001b[0m= \u001b[96m\u001b[1mphi\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[33m[\u001b[39m \u001b[95mtrue\u001b[39m\u001b[0m, \u001b[91m%L49\u001b[39m \u001b[33m]\u001b[39m\u001b[0m, \u001b[33m[\u001b[39m \u001b[95mfalse\u001b[39m\u001b[0m, \u001b[91m%top\u001b[39m \u001b[33m]\u001b[39m\n",
      "\u001b[90m; └\u001b[39m\n",
      "\u001b[90m;  @ In[64]:4 within `sumofsins1'\u001b[39m\n",
      "  \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[0m%3\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L31\u001b[39m\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L28\u001b[39m\n",
      "\n",
      "\u001b[91mL28:\u001b[39m                                              \u001b[90m; preds = %L11\u001b[39m\n",
      "\u001b[90m; ┌ @ float.jl:326 within `+'\u001b[39m\n",
      "   \u001b[0m%5 \u001b[0m= \u001b[96m\u001b[1mbitcast\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%.sroa.014.0 \u001b[95mto\u001b[39m \u001b[36mdouble\u001b[39m\n",
      "\u001b[90m; └\u001b[39m\n",
      "\u001b[90m;  @ In[64]:3 within `sumofsins1'\u001b[39m\n",
      "\u001b[90m; ┌ @ range.jl:670 within `iterate'\u001b[39m\n",
      "   \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mlabel\u001b[39m \u001b[91m%L39\u001b[39m\n",
      "\n",
      "\u001b[91mL31:\u001b[39m                                              \u001b[90m; preds = %L11\u001b[39m\n",
      "\u001b[90m; └\u001b[39m\n",
      "\u001b[90m;  @ In[64]:4 within `sumofsins1'\u001b[39m\n",
      "  \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[0m%4\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L37\u001b[39m\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L33\u001b[39m\n",
      "\n",
      "\u001b[91mL33:\u001b[39m                                              \u001b[90m; preds = %L31\u001b[39m\n",
      "\u001b[90m; ┌ @ promotion.jl:321 within `+'\u001b[39m\n",
      "\u001b[90m; │┌ @ promotion.jl:292 within `promote'\u001b[39m\n",
      "\u001b[90m; ││┌ @ promotion.jl:269 within `_promote'\u001b[39m\n",
      "\u001b[90m; │││┌ @ number.jl:7 within `convert'\u001b[39m\n",
      "\u001b[90m; ││││┌ @ float.jl:94 within `Float64'\u001b[39m\n",
      "       \u001b[0m%6 \u001b[0m= \u001b[96m\u001b[1msitofp\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%.sroa.014.0 \u001b[95mto\u001b[39m \u001b[36mdouble\u001b[39m\n",
      "\u001b[90m; └└└└└\u001b[39m\n",
      "\u001b[90m;  @ In[64]:3 within `sumofsins1'\u001b[39m\n",
      "\u001b[90m; ┌ @ range.jl:670 within `iterate'\u001b[39m\n",
      "   \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mlabel\u001b[39m \u001b[91m%L39\u001b[39m\n",
      "\n",
      "\u001b[91mL37:\u001b[39m                                              \u001b[90m; preds = %L31\u001b[39m\n",
      "\u001b[90m; └\u001b[39m\n",
      "\u001b[90m;  @ In[64]:4 within `sumofsins1'\u001b[39m\n",
      "  \u001b[96m\u001b[1mcall\u001b[22m\u001b[39m \u001b[36mvoid\u001b[39m \u001b[93m@jl_throw\u001b[39m\u001b[33m(\u001b[39m\u001b[33m{\u001b[39m\u001b[33m}\u001b[39m\u001b[0m* \u001b[95minttoptr\u001b[39m \u001b[33m(\u001b[39m\u001b[36mi64\u001b[39m \u001b[33m4515526624\u001b[39m \u001b[95mto\u001b[39m \u001b[33m{\u001b[39m\u001b[33m}\u001b[39m\u001b[0m*\u001b[33m)\u001b[39m\u001b[33m)\u001b[39m\n",
      "  \u001b[96m\u001b[1munreachable\u001b[22m\u001b[39m\n",
      "\n",
      "\u001b[91mL39:\u001b[39m                                              \u001b[90m; preds = %L33, %L28\u001b[39m\n",
      "  \u001b[0m%value_phi3.in \u001b[0m= \u001b[96m\u001b[1mphi\u001b[22m\u001b[39m \u001b[36mdouble\u001b[39m \u001b[33m[\u001b[39m \u001b[0m%5\u001b[0m, \u001b[91m%L28\u001b[39m \u001b[33m]\u001b[39m\u001b[0m, \u001b[33m[\u001b[39m \u001b[0m%6\u001b[0m, \u001b[91m%L33\u001b[39m \u001b[33m]\u001b[39m\n",
      "\u001b[90m;  @ In[64] within `sumofsins1'\u001b[39m\n",
      "  \u001b[0m%value_phi3 \u001b[0m= \u001b[96m\u001b[1mfadd\u001b[22m\u001b[39m \u001b[36mdouble\u001b[39m \u001b[0m%value_phi3.in\u001b[0m, \u001b[33m0xBFD05AC910FF4C6C\u001b[39m\n",
      "\u001b[90m;  @ In[64]:4 within `sumofsins1'\u001b[39m\n",
      "\u001b[90m; ┌ @ range.jl:674 within `iterate'\u001b[39m\n",
      "\u001b[90m; │┌ @ promotion.jl:410 within `=='\u001b[39m\n",
      "    \u001b[0m%.not.not \u001b[0m= \u001b[96m\u001b[1micmp\u001b[22m\u001b[39m \u001b[96m\u001b[1meq\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%value_phi2\u001b[0m, \u001b[0m%2\n",
      "\u001b[90m; └└\u001b[39m\n",
      "  \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[0m%.not.not\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%union_move\u001b[39m\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L49\u001b[39m\n",
      "\n",
      "\u001b[91mL49:\u001b[39m                                              \u001b[90m; preds = %L39\u001b[39m\n",
      "\u001b[90m; ┌ @ range.jl:674 within `iterate'\u001b[39m\n",
      "   \u001b[0m%7 \u001b[0m= \u001b[96m\u001b[1madd\u001b[22m\u001b[39m \u001b[95mnuw\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%value_phi2\u001b[0m, \u001b[33m1\u001b[39m\n",
      "\u001b[90m; └\u001b[39m\n",
      "\u001b[90m;  @ In[64]:3 within `sumofsins1'\u001b[39m\n",
      "\u001b[90m; ┌ @ range.jl:670 within `iterate'\u001b[39m\n",
      "   \u001b[0m%8 \u001b[0m= \u001b[96m\u001b[1mbitcast\u001b[22m\u001b[39m \u001b[36mdouble\u001b[39m \u001b[0m%value_phi3 \u001b[95mto\u001b[39m \u001b[36mi64\u001b[39m\n",
      "   \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mlabel\u001b[39m \u001b[91m%L11\u001b[39m\n",
      "\n",
      "\u001b[91mpost_union_move:\u001b[39m                                  \u001b[90m; preds = %union_move8, %union_move\u001b[39m\n",
      "   \u001b[0m%tindex_phi622 \u001b[0m= \u001b[96m\u001b[1mphi\u001b[22m\u001b[39m \u001b[36mi8\u001b[39m \u001b[33m[\u001b[39m \u001b[33m2\u001b[39m\u001b[0m, \u001b[91m%union_move8\u001b[39m \u001b[33m]\u001b[39m\u001b[0m, \u001b[33m[\u001b[39m \u001b[33m1\u001b[39m\u001b[0m, \u001b[91m%union_move\u001b[39m \u001b[33m]\u001b[39m\n",
      "\u001b[90m; └\u001b[39m\n",
      "\u001b[90m;  @ In[64]:6 within `sumofsins1'\u001b[39m\n",
      "  \u001b[0m%9 \u001b[0m= \u001b[96m\u001b[1minsertvalue\u001b[22m\u001b[39m \u001b[33m{\u001b[39m \u001b[33m{\u001b[39m\u001b[33m}\u001b[39m\u001b[0m*\u001b[0m, \u001b[36mi8\u001b[39m \u001b[33m}\u001b[39m \u001b[33m{\u001b[39m \u001b[33m{\u001b[39m\u001b[33m}\u001b[39m\u001b[0m* \u001b[95mnull\u001b[39m\u001b[0m, \u001b[36mi8\u001b[39m \u001b[95mundef\u001b[39m \u001b[33m}\u001b[39m\u001b[0m, \u001b[36mi8\u001b[39m \u001b[0m%tindex_phi622\u001b[0m, \u001b[33m1\u001b[39m\n",
      "  \u001b[96m\u001b[1mret\u001b[22m\u001b[39m \u001b[33m{\u001b[39m \u001b[33m{\u001b[39m\u001b[33m}\u001b[39m\u001b[0m*\u001b[0m, \u001b[36mi8\u001b[39m \u001b[33m}\u001b[39m \u001b[0m%9\n",
      "\n",
      "\u001b[91munion_move:\u001b[39m                                       \u001b[90m; preds = %L39\u001b[39m\n",
      "  \u001b[0m%10 \u001b[0m= \u001b[96m\u001b[1mbitcast\u001b[22m\u001b[39m \u001b[33m[\u001b[39m\u001b[33m8\u001b[39m \u001b[0mx \u001b[36mi8\u001b[39m\u001b[33m]\u001b[39m\u001b[0m* \u001b[0m%0 \u001b[95mto\u001b[39m \u001b[36mdouble\u001b[39m\u001b[0m*\n",
      "  \u001b[96m\u001b[1mstore\u001b[22m\u001b[39m \u001b[36mdouble\u001b[39m \u001b[0m%value_phi3\u001b[0m, \u001b[36mdouble\u001b[39m\u001b[0m* \u001b[0m%10\u001b[0m, \u001b[95malign\u001b[39m \u001b[33m8\u001b[39m\n",
      "  \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mlabel\u001b[39m \u001b[91m%post_union_move\u001b[39m\n",
      "\n",
      "\u001b[91munion_move8:\u001b[39m                                      \u001b[90m; preds = %top\u001b[39m\n",
      "  \u001b[0m%11 \u001b[0m= \u001b[96m\u001b[1mbitcast\u001b[22m\u001b[39m \u001b[33m[\u001b[39m\u001b[33m8\u001b[39m \u001b[0mx \u001b[36mi8\u001b[39m\u001b[33m]\u001b[39m\u001b[0m* \u001b[0m%0 \u001b[95mto\u001b[39m \u001b[36mi64\u001b[39m\u001b[0m*\n",
      "  \u001b[96m\u001b[1mstore\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[33m0\u001b[39m\u001b[0m, \u001b[36mi64\u001b[39m\u001b[0m* \u001b[0m%11\u001b[0m, \u001b[95malign\u001b[39m \u001b[33m8\u001b[39m\n",
      "  \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mlabel\u001b[39m \u001b[91m%post_union_move\u001b[39m\n",
      "\u001b[33m}\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "@code_llvm(sumofsins1(100))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[90m;  @ In[67]:1 within `sumofsins2'\u001b[39m\n",
      "\u001b[95mdefine\u001b[39m \u001b[36mdouble\u001b[39m \u001b[93m@julia_sumofsins2_7316\u001b[39m\u001b[33m(\u001b[39m\u001b[36mi64\u001b[39m \u001b[95msignext\u001b[39m \u001b[0m%0\u001b[33m)\u001b[39m \u001b[33m{\u001b[39m\n",
      "\u001b[91mtop:\u001b[39m\n",
      "\u001b[90m;  @ In[67]:3 within `sumofsins2'\u001b[39m\n",
      "\u001b[90m; ┌ @ range.jl:5 within `Colon'\u001b[39m\n",
      "\u001b[90m; │┌ @ range.jl:287 within `UnitRange'\u001b[39m\n",
      "\u001b[90m; ││┌ @ range.jl:292 within `unitrange_last'\u001b[39m\n",
      "     \u001b[0m%.inv \u001b[0m= \u001b[96m\u001b[1micmp\u001b[22m\u001b[39m \u001b[96m\u001b[1msgt\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%0\u001b[0m, \u001b[33m0\u001b[39m\n",
      "     \u001b[0m%1 \u001b[0m= \u001b[96m\u001b[1mselect\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[0m%.inv\u001b[0m, \u001b[36mi64\u001b[39m \u001b[0m%0\u001b[0m, \u001b[36mi64\u001b[39m \u001b[33m0\u001b[39m\n",
      "\u001b[90m; └└└\u001b[39m\n",
      "  \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[0m%.inv\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L11\u001b[39m\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L37\u001b[39m\n",
      "\n",
      "\u001b[91mL11:\u001b[39m                                              \u001b[90m; preds = %L11, %top\u001b[39m\n",
      "  \u001b[0m%value_phi2 \u001b[0m= \u001b[96m\u001b[1mphi\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[33m[\u001b[39m \u001b[0m%3\u001b[0m, \u001b[91m%L11\u001b[39m \u001b[33m]\u001b[39m\u001b[0m, \u001b[33m[\u001b[39m \u001b[33m1\u001b[39m\u001b[0m, \u001b[91m%top\u001b[39m \u001b[33m]\u001b[39m\n",
      "  \u001b[0m%value_phi3 \u001b[0m= \u001b[96m\u001b[1mphi\u001b[22m\u001b[39m \u001b[36mdouble\u001b[39m \u001b[33m[\u001b[39m \u001b[0m%2\u001b[0m, \u001b[91m%L11\u001b[39m \u001b[33m]\u001b[39m\u001b[0m, \u001b[33m[\u001b[39m \u001b[33m0.000000e+00\u001b[39m\u001b[0m, \u001b[91m%top\u001b[39m \u001b[33m]\u001b[39m\n",
      "\u001b[90m;  @ In[67]:4 within `sumofsins2'\u001b[39m\n",
      "\u001b[90m; ┌ @ float.jl:326 within `+'\u001b[39m\n",
      "   \u001b[0m%2 \u001b[0m= \u001b[96m\u001b[1mfadd\u001b[22m\u001b[39m \u001b[36mdouble\u001b[39m \u001b[0m%value_phi3\u001b[0m, \u001b[33m0xBFD05AC910FF4C6C\u001b[39m\n",
      "\u001b[90m; └\u001b[39m\n",
      "\u001b[90m; ┌ @ range.jl:674 within `iterate'\u001b[39m\n",
      "\u001b[90m; │┌ @ promotion.jl:410 within `=='\u001b[39m\n",
      "    \u001b[0m%.not.not \u001b[0m= \u001b[96m\u001b[1micmp\u001b[22m\u001b[39m \u001b[96m\u001b[1meq\u001b[22m\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%value_phi2\u001b[0m, \u001b[0m%1\n",
      "\u001b[90m; │└\u001b[39m\n",
      "   \u001b[0m%3 \u001b[0m= \u001b[96m\u001b[1madd\u001b[22m\u001b[39m \u001b[95mnuw\u001b[39m \u001b[36mi64\u001b[39m \u001b[0m%value_phi2\u001b[0m, \u001b[33m1\u001b[39m\n",
      "\u001b[90m; └\u001b[39m\n",
      "  \u001b[96m\u001b[1mbr\u001b[22m\u001b[39m \u001b[36mi1\u001b[39m \u001b[0m%.not.not\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L37\u001b[39m\u001b[0m, \u001b[36mlabel\u001b[39m \u001b[91m%L11\u001b[39m\n",
      "\n",
      "\u001b[91mL37:\u001b[39m                                              \u001b[90m; preds = %L11, %top\u001b[39m\n",
      "  \u001b[0m%value_phi6 \u001b[0m= \u001b[96m\u001b[1mphi\u001b[22m\u001b[39m \u001b[36mdouble\u001b[39m \u001b[33m[\u001b[39m \u001b[33m0.000000e+00\u001b[39m\u001b[0m, \u001b[91m%top\u001b[39m \u001b[33m]\u001b[39m\u001b[0m, \u001b[33m[\u001b[39m \u001b[0m%2\u001b[0m, \u001b[91m%L11\u001b[39m \u001b[33m]\u001b[39m\n",
      "\u001b[90m;  @ In[67]:6 within `sumofsins2'\u001b[39m\n",
      "  \u001b[96m\u001b[1mret\u001b[22m\u001b[39m \u001b[36mdouble\u001b[39m \u001b[0m%value_phi6\n",
      "\u001b[33m}\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "@code_llvm(sumofsins2(100))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Julia 1.6.2",
   "language": "julia",
   "name": "julia-1.6"
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