math.ln

export function ln(x: number): number

import { ln } from "#fune/math.ln@^1";

// Derived from fdlibm e_log.c.
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
// Developed at SunSoft, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.

// fdlibm's split of ln 2: the high part has trailing zero bits, so e * LN2_HI
// is exact for any exponent a double can have.
const LN2_HI = 6.93147180369123816490e-01;
const LN2_LO = 1.90821492927058770002e-10;
const SQRT2 = 1.4142135623730951;

/**
 * Natural logarithm using only +, -, * and /, which IEEE 754 defines exactly.
 *
 * Math.log is whatever the platform's maths library does, and its last bit is
 * allowed to differ between JavaScript engines, CPython builds and Rust
 * targets. Rounding the result afterwards does not hide that when a value sits
 * next to a rounding boundary, so capabilities that must agree across
 * languages (a log scale, a colour conversion) use this instead.
 *
 * x = m * 2^e with m in (sqrt(1/2), sqrt(2)], found by exact halving and
 * doubling; then ln m = 2 atanh(s) with s = (m - 1) / (m + 1), |s| < 0.172,
 * summed as an odd series in s to well below an ulp.
 */
export function ln(x: number): number {
  if (typeof x !== "number" || !Number.isFinite(x) || x <= 0) {
    throw new RangeError(`ln is only defined for finite numbers greater than zero, received ${x}`);
  }
  let m = x;
  let e = 0;
  while (m >= 2) {
    m /= 2;
    e += 1;
  }
  while (m < 1) {
    m *= 2;
    e -= 1;
  }
  if (m > SQRT2) {
    m /= 2;
    e += 1;
  }
  const f = m - 1;
  const s = f / (2 + f);
  const z = s * s;
  let p = 1 / 25;
  p = 1 / 23 + z * p;
  p = 1 / 21 + z * p;
  p = 1 / 19 + z * p;
  p = 1 / 17 + z * p;
  p = 1 / 15 + z * p;
  p = 1 / 13 + z * p;
  p = 1 / 11 + z * p;
  p = 1 / 9 + z * p;
  p = 1 / 7 + z * p;
  p = 1 / 5 + z * p;
  p = 1 / 3 + z * p;
  const lnm = 2 * s + 2 * s * (z * p);
  return e * LN2_HI + (e * LN2_LO + lnm);
}

Install

fune build

With that line in your source, in a TypeScript project (language typescript in fune.project), fune build resolves it and nothing else, pins them in fune.lock, downloads only the TypeScript package of each, and builds the code above into your project’s .fune/build, one readable file per capability with a header linking back here. Or pin a range in fune.project and build in one step:

fune add math.ln

The manifest, vectors and README with only the TypeScript implementation. Install it without the registry with fune add ./math.ln-1.0.1-typescript.fune, or fetch it from a terminal with fune pull math.ln@1.0.1:typescript.

The whole function, every language, is one file too: math.ln-1.0.1.fune, 10,563 bytes, sha256 fcc649f7bc35683193541d1555c0148e1698113437f262b6ec72d5868216ca15. It installs into a project of any language.

// fune: before math.ln

// fune: after math.ln

// fune: step math.ln after <n|label>

More from the author

## Method

1. Write x = m x 2^e with m in (sqrt(1/2), sqrt(2)], by halving or doubling, which is exact for every double, subnormals included. 2. With s = (m - 1) / (m + 1), |s| < 0.172, ln m = 2 atanh s = 2s (1 + s²/3 + s⁴/5 + ... + s²⁴/25), summed by Horner's rule. 3. ln x = e ln 2 + ln m, with ln 2 split into a high part whose trailing bits are zero (so e x high is exact) and a low correction, as in fdlibm.

The result is within 2 units in the last place of the true logarithm over the whole positive range (checked against a correctly rounded reference), and bit-identical in all three languages because every step is an IEEE operation performed in the same order. `ln(1)` is exactly 0.

Zero, negative numbers, infinity and NaN are errors.

Sources: W. J. Cody and W. Waite, *Software Manual for the Elementary Functions* (1980); Sun Microsystems fdlibm, `e_log.c` (the ln 2 split).

## Notices

Derived from fdlibm e_log.c. Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. Developed at SunSoft, a Sun Microsystems, Inc. business. Permission to use, copy, modify, and distribute this software is freely granted, provided that this notice is preserved.

The same notice heads each implementation file.

1.0.1 adds its attribution notices (NOTICE). The code is unchanged apart from that notice at the top of each implementation file; the tests are unchanged.

Files