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title: How Does Go Compute a Logarithm
description: Source Diving and Math
date: 2017-02-06
author: Danny Hermes (
tags: Math, Programming, Logarithm, Approximation, Remez
slug: golang-and-log-x
comments: true
use_twitter_card: true
use_open_graph: true
use_schema_org: true
twitter_site: @bossylobster
twitter_creator: @bossylobster
social_image: images/remez_equioscillating_error.png
github_slug: templated_content/2017-02-06-golang-and-log-x.template
About a year ago, I was reading the Go [source][1] for computing
{{ get_katex("\\log(x)") }} and was very surprised by the
implementation.[ref]Why would anyone look in this source? I was
trying to explain to a student that a computer would compute
this with minimal error when I realized I didn't know **how**.[/ref]
Even three years into a PhD in Applied Math (i.e. numerics), I still
managed to learn something by diving in and trying to understand what
was going on.
### (Almost-)Correctness
Really I just wanted a chance to share a neat error plot:
<div markdown="1" style="text-align: center;">
![Equioscillating Error](/images/remez_equioscillating_error.png)
The high-level:
- The computation {{ get_katex("\\log(x)") }} can be
reduced to computing a related function {{ get_katex("R(s)") }}
(where the value {{ get_katex("s") }} can be obtained from
{{ get_katex("x") }} on a computer in some straightforward way)
- {{ get_katex("R(s)") }} can be approximated by a function that
we can actually compute with simple floating point operations:
{{ get_katex("f(s)") }}
- The error {{ get_katex("R(s) - f(s)") }} is ["equioscillating"][5],
i.e. it minimizes the maximum error, a so-called
[minimax][4] solution
If you're still interested in some mathematical details, stick
around. If not, I hope you enjoyed the pretty picture.
### Doing the Obvious
The "obvious" idea that comes to mind is to use a [Taylor series][2]
approximation. But {{ get_katex("\\log(x)") }} is not defined at
{{ get_katex("x = 0,") }} so a Taylor series of a related function is
typically used[ref]This series is typically learned and forgotten by
those who have made it through "Calc II".[/ref]:
{{ get_katex("\\log\\left(1 + x\\right) = x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\frac{x^4}{4} + \\cdots", blockquote=True) }}
But, the implementation doesn't do the obvious thing. Instead
it breaks down the argument into an exponent and a fractional part
{{ get_katex("x = 2^k \\left(\\frac{1 + s}{1 - s}\\right)", blockquote=True) }}
which transforms the computation to:
{{ get_katex("\\log(x) = k \\log(2) + \\left[\\log\\left(1 + s\\right) - \\log\\left(1 - s\\right)\\right].", blockquote=True) }}
The first part {{ get_katex("k \\log(2)") }} can be handled in maximal
precision for the limited set of integer values that {{ get_katex("k") }}
can be.
### Finding a Function to Approximate
For the other part
{{ get_katex("\\begin{aligned}\\log\\left(1 + s\\right) - \\log\\left(1 - s\\right) &= 2 s + \\frac{2}{3} s^3 + \\frac{2}{5} s^5 + \\cdots \\\\ &= 2 s + s R(s). \\end{aligned}", blockquote=True) }}
From here, I expected a polynomial approximation that just takes the first
few terms in {{ get_katex("R(s)") }}.
However, coefficients that are close (but not equal) are given:
{{ get_katex("R(z) \\approx L_1 s^2 + L_2 s^4 + \\cdots + L_6 s^{12} + L_7 s^{14}", blockquote=True) }}
L1 = 6.666666666666735130e-01 /* 3FE55555 55555593 */
L2 = 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
L3 = 2.857142874366239149e-01 /* 3FD24924 94229359 */
L4 = 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
L5 = 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
L6 = 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
L7 = 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
For example:
{{ get_katex("L_1 = 2^{-1} \\cdot 1.5555555555593_{16} = \\frac{2}{3} + 2^{-53} \\cdot \\frac{185}{3} \\approx \\frac{2}{3}.", blockquote=True) }}
### Not Go, Sun
You may have noted my typo above: I used {{ get_katex("R(z)") }}
when I should've used {{ get_katex("R(s)") }}. This typo is
directly from the source; not my copy-paste error, but a
copy-paste error in the original source.
The original authors aren't the Golang devs:
// The original C code, the long comment, and the constants
// below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
// and came with this notice. The go code is a simpler
// version of the original C.
It seems they copied **everything**, even the typos. Another
typo can be found when the description mentions
"a special Reme algorithm" used to compute the coefficients
{{ get_katex("L_j") }}.
This typo is a bit more egregious: it actually uses a
[**Remez** algorithm][3].
### Remez Algorithm
This algorithm takes a polynomial approximation like
{{ get_katex("R(s) \\approx \\frac{2}{3} s^2 + \\frac{2}{5} s^4 + \\cdots + \\frac{2}{15} s^{14}", blockquote=True) }}
and systematically modifies the coefficients until the error is
<div markdown="1" style="text-align: center;">
![Equioscillating Error](/images/remez_equioscillating_error.png)
### Is That All You've Got?
I hope to follow this post up with [some code][6] to actually compute
the coefficients that give an equioscillating error (and to show some
coefficients that do slightly better).
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