Average Error: 39.4 → 0.3
Time: 3.9s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000003968673248033738332196662668138742:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(1 + x\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.000003968673248033738332196662668138742:\\
\;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\

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

\end{array}
double f(double x) {
        double r104285 = 1.0;
        double r104286 = x;
        double r104287 = r104285 + r104286;
        double r104288 = log(r104287);
        return r104288;
}
↓
double f(double x) {
        double r104289 = 1.0;
        double r104290 = x;
        double r104291 = r104289 + r104290;
        double r104292 = 1.000003968673248;
        bool r104293 = r104291 <= r104292;
        double r104294 = log(r104289);
        double r104295 = 0.5;
        double r104296 = 2.0;
        double r104297 = pow(r104290, r104296);
        double r104298 = pow(r104289, r104296);
        double r104299 = r104297 / r104298;
        double r104300 = r104295 * r104299;
        double r104301 = r104294 - r104300;
        double r104302 = fma(r104290, r104289, r104301);
        double r104303 = log(r104291);
        double r104304 = r104295 * r104303;
        double r104305 = sqrt(r104291);
        double r104306 = log(r104305);
        double r104307 = r104304 + r104306;
        double r104308 = r104293 ? r104302 : r104307;
        return r104308;
}

Error

Bits error versus x

Target

Original39.4
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) < 1.000003968673248

    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. Simplified0.3

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

    if 1.000003968673248 < (+ 1.0 x)

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied log-prod0.2

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

    6. Applied pow1/20.2

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

    7. Applied log-pow0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000003968673248033738332196662668138742:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(1 + x\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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)))