ln(1 + x)

Average Error: 39.9 → 0.3

Time: 12.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\log \left(1 + x\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000464934527766303062890074215829372:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{\frac{-1}{2}}{1 \cdot 1}, x, 1\right), \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

C

double f(double x) {
        double r57274 = 1.0;
        double r57275 = x;
        double r57276 = r57274 + r57275;
        double r57277 = log(r57276);
        return r57277;
}

↓

double f(double x) {
        double r57278 = 1.0;
        double r57279 = x;
        double r57280 = r57278 + r57279;
        double r57281 = 1.0000004649345278;
        bool r57282 = r57280 <= r57281;
        double r57283 = -0.5;
        double r57284 = r57278 * r57278;
        double r57285 = r57283 / r57284;
        double r57286 = fma(r57285, r57279, r57278);
        double r57287 = log(r57278);
        double r57288 = fma(r57279, r57286, r57287);
        double r57289 = log(r57280);
        double r57290 = r57282 ? r57288 : r57289;
        return r57290;
}

Error

Bits error versus x

Target

Original	39.9
Target	0.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (+ 1.0 x) < 1.0000004649345278

   1. Initial program 59.2

      \[\log \left(1 + x\right)\]

   2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\]

   3. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{\frac{-1}{2}}{1 \cdot 1}, x, 1\right), \log 1\right)}\]

   if 1.0000004649345278 < (+ 1.0 x)

   1. Initial program 0.2

      \[\log \left(1 + x\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000464934527766303062890074215829372:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{\frac{-1}{2}}{1 \cdot 1}, x, 1\right), \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

  :herbie-target
  (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1)))

  (log (+ 1 x)))