Average Error: 38.7 → 0.3
Time: 15.3s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000005155942339740704483119770884514:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.000000005155942339740704483119770884514:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\

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

\end{array}
double f(double x) {
        double r34946 = 1.0;
        double r34947 = x;
        double r34948 = r34946 + r34947;
        double r34949 = log(r34948);
        return r34949;
}
↓
double f(double x) {
        double r34950 = 1.0;
        double r34951 = x;
        double r34952 = r34950 + r34951;
        double r34953 = 1.0000000051559423;
        bool r34954 = r34952 <= r34953;
        double r34955 = -0.5;
        double r34956 = 2.0;
        double r34957 = pow(r34951, r34956);
        double r34958 = pow(r34950, r34956);
        double r34959 = r34957 / r34958;
        double r34960 = log(r34950);
        double r34961 = fma(r34950, r34951, r34960);
        double r34962 = fma(r34955, r34959, r34961);
        double r34963 = log(r34952);
        double r34964 = r34954 ? r34962 : r34963;
        return r34964;
}

Error

Bits error versus x

Target

Original38.7
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) < 1.0000000051559423

    1. Initial program 59.2

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

    3. Applied expm1-log1p-u59.2

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(1 + x\right)\right)\right)}\]

    4. Taylor expanded around 0 0.3

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

    5. Simplified0.3

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

    if 1.0000000051559423 < (+ 1.0 x)

    1. Initial program 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000005155942339740704483119770884514:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))