Average Error: 38.5 → 0.3
Time: 3.8s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.0000000771282975:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.0000000771282975:\\
\;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\

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

\end{array}
double f(double x) {
        double r74114 = 1.0;
        double r74115 = x;
        double r74116 = r74114 + r74115;
        double r74117 = log(r74116);
        return r74117;
}
↓
double f(double x) {
        double r74118 = 1.0;
        double r74119 = x;
        double r74120 = r74118 + r74119;
        double r74121 = 1.0000000771282975;
        bool r74122 = r74120 <= r74121;
        double r74123 = log(r74118);
        double r74124 = 0.5;
        double r74125 = 2.0;
        double r74126 = pow(r74119, r74125);
        double r74127 = pow(r74118, r74125);
        double r74128 = r74126 / r74127;
        double r74129 = r74124 * r74128;
        double r74130 = r74123 - r74129;
        double r74131 = fma(r74119, r74118, r74130);
        double r74132 = sqrt(r74120);
        double r74133 = log(r74132);
        double r74134 = r74133 + r74133;
        double r74135 = r74122 ? r74131 : r74134;
        return r74135;
}

Error

Bits error versus x

Target

Original38.5
Target0.3
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) < 1.0000000771282975

    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 1.0000000771282975 < (+ 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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.0000000771282975:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))