Average Error: 39.3 → 0.3
Time: 3.9s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.0000000084552092:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \log \left(\sqrt{1 + x}\right)\right) \cdot \log \left(1 + x\right)}\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.0000000084552092:\\
\;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(2 \cdot \log \left(\sqrt{1 + x}\right)\right) \cdot \log \left(1 + x\right)}\\

\end{array}
double f(double x) {
        double r100305 = 1.0;
        double r100306 = x;
        double r100307 = r100305 + r100306;
        double r100308 = log(r100307);
        return r100308;
}
↓
double f(double x) {
        double r100309 = 1.0;
        double r100310 = x;
        double r100311 = r100309 + r100310;
        double r100312 = 1.0000000084552092;
        bool r100313 = r100311 <= r100312;
        double r100314 = log(r100309);
        double r100315 = 0.5;
        double r100316 = 2.0;
        double r100317 = pow(r100310, r100316);
        double r100318 = pow(r100309, r100316);
        double r100319 = r100317 / r100318;
        double r100320 = r100315 * r100319;
        double r100321 = r100314 - r100320;
        double r100322 = fma(r100310, r100309, r100321);
        double r100323 = sqrt(r100311);
        double r100324 = log(r100323);
        double r100325 = r100316 * r100324;
        double r100326 = log(r100311);
        double r100327 = r100325 * r100326;
        double r100328 = sqrt(r100327);
        double r100329 = r100313 ? r100322 : r100328;
        return r100329;
}

Error

Bits error versus x

Target

Original39.3
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) < 1.0000000084552092

    1. Initial program 59.4

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

    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

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

    if 1.0000000084552092 < (+ 1.0 x)

    1. Initial program 0.3

      \[\log \left(1 + x\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.8

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + x\right)} \cdot \sqrt{\log \left(1 + x\right)}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.8

      \[\leadsto \sqrt{\log \left(1 + x\right)} \cdot \sqrt{\log \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x}\right)}}\]

    6. Applied log-prod0.8

      \[\leadsto \sqrt{\log \left(1 + x\right)} \cdot \sqrt{\color{blue}{\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)}}\]
    7. Using strategy rm

    8. Applied sqrt-unprod0.3

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + x\right) \cdot \left(\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\right)}}\]

    9. Simplified0.3

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \log \left(\sqrt{1 + x}\right)\right) \cdot \log \left(1 + x\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.0000000084552092:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \log \left(\sqrt{1 + x}\right)\right) \cdot \log \left(1 + x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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)))