Average Error: 38.9 → 0.3
Time: 25.2s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000014348698229582623753231018781662:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \frac{x}{1} \cdot \frac{x}{1}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.000000014348698229582623753231018781662:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \frac{x}{1} \cdot \frac{x}{1}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(1 + x\right)\\

\end{array}
double f(double x) {
        double r4470600 = 1.0;
        double r4470601 = x;
        double r4470602 = r4470600 + r4470601;
        double r4470603 = log(r4470602);
        return r4470603;
}
↓
double f(double x) {
        double r4470604 = 1.0;
        double r4470605 = x;
        double r4470606 = r4470604 + r4470605;
        double r4470607 = 1.0000000143486982;
        bool r4470608 = r4470606 <= r4470607;
        double r4470609 = -0.5;
        double r4470610 = r4470605 / r4470604;
        double r4470611 = r4470610 * r4470610;
        double r4470612 = log(r4470604);
        double r4470613 = fma(r4470604, r4470605, r4470612);
        double r4470614 = fma(r4470609, r4470611, r4470613);
        double r4470615 = log(r4470606);
        double r4470616 = r4470608 ? r4470614 : r4470615;
        return r4470616;
}

Error

Bits error versus x

Target

Original38.9
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ 1.0 x) < 1.0000000143486982

    1. Initial program 59.1

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

    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. Simplified0.3

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

    if 1.0000000143486982 < (+ 1.0 x)

    1. Initial program 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000014348698229582623753231018781662:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \frac{x}{1} \cdot \frac{x}{1}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))