Average Error: 39.2 → 0.3
Time: 11.7s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000005236653777274113963358104228973:\\ \;\;\;\;\mathsf{fma}\left(x, 1 - x \cdot 0.5, \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.000000005236653777274113963358104228973:\\
\;\;\;\;\mathsf{fma}\left(x, 1 - x \cdot 0.5, \log 1\right)\\

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

\end{array}
double f(double x) {
        double r45162 = 1.0;
        double r45163 = x;
        double r45164 = r45162 + r45163;
        double r45165 = log(r45164);
        return r45165;
}
↓
double f(double x) {
        double r45166 = 1.0;
        double r45167 = x;
        double r45168 = r45166 + r45167;
        double r45169 = 1.0000000052366538;
        bool r45170 = r45168 <= r45169;
        double r45171 = 0.5;
        double r45172 = r45167 * r45171;
        double r45173 = r45166 - r45172;
        double r45174 = log(r45166);
        double r45175 = fma(r45167, r45173, r45174);
        double r45176 = log(r45168);
        double r45177 = r45170 ? r45175 : r45176;
        return r45177;
}

Error

Bits error versus x

Target

Original39.2
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) < 1.0000000052366538

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

    4. Taylor expanded around 0 0.4

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

    5. Simplified0.4

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

    if 1.0000000052366538 < (+ 1.0 x)

    1. Initial program 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000005236653777274113963358104228973:\\ \;\;\;\;\mathsf{fma}\left(x, 1 - x \cdot 0.5, \log 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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)))