Average Error: 38.8 → 0.3
Time: 22.7s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + 1 \le 1.000001508099978009269648282497655600309:\\ \;\;\;\;\mathsf{fma}\left(1 - 0.5 \cdot x, x, \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;x + 1 \le 1.000001508099978009269648282497655600309:\\
\;\;\;\;\mathsf{fma}\left(1 - 0.5 \cdot x, x, \log 1\right)\\

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

\end{array}
double f(double x) {
        double r3043565 = 1.0;
        double r3043566 = x;
        double r3043567 = r3043565 + r3043566;
        double r3043568 = log(r3043567);
        return r3043568;
}
↓
double f(double x) {
        double r3043569 = x;
        double r3043570 = 1.0;
        double r3043571 = r3043569 + r3043570;
        double r3043572 = 1.000001508099978;
        bool r3043573 = r3043571 <= r3043572;
        double r3043574 = 0.5;
        double r3043575 = r3043574 * r3043569;
        double r3043576 = r3043570 - r3043575;
        double r3043577 = log(r3043570);
        double r3043578 = fma(r3043576, r3043569, r3043577);
        double r3043579 = sqrt(r3043571);
        double r3043580 = log(r3043579);
        double r3043581 = r3043580 + r3043580;
        double r3043582 = r3043573 ? r3043578 : r3043581;
        return r3043582;
}

Error

Bits error versus x

Target

Original38.8
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) < 1.000001508099978

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

    4. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot {x}^{2}}\]

    5. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - 0.5 \cdot x, x, \log 1\right)}\]

    if 1.000001508099978 < (+ 1.0 x)

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied log-prod0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \le 1.000001508099978009269648282497655600309:\\ \;\;\;\;\mathsf{fma}\left(1 - 0.5 \cdot x, x, \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))