Average Error: 39.3 → 0.3
Time: 3.8s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.0000000084552092:\\ \;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \log \left(\sqrt{1 + x}\right)\right) \cdot \log \left(1 + x\right)}\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.0000000084552092:\\
\;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\\

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

\end{array}
double f(double x) {
        double r75193 = 1.0;
        double r75194 = x;
        double r75195 = r75193 + r75194;
        double r75196 = log(r75195);
        return r75196;
}
↓
double f(double x) {
        double r75197 = 1.0;
        double r75198 = x;
        double r75199 = r75197 + r75198;
        double r75200 = 1.0000000084552092;
        bool r75201 = r75199 <= r75200;
        double r75202 = r75197 * r75198;
        double r75203 = log(r75197);
        double r75204 = r75202 + r75203;
        double r75205 = 0.5;
        double r75206 = 2.0;
        double r75207 = pow(r75198, r75206);
        double r75208 = pow(r75197, r75206);
        double r75209 = r75207 / r75208;
        double r75210 = r75205 * r75209;
        double r75211 = r75204 - r75210;
        double r75212 = sqrt(r75199);
        double r75213 = log(r75212);
        double r75214 = r75206 * r75213;
        double r75215 = log(r75199);
        double r75216 = r75214 * r75215;
        double r75217 = sqrt(r75216);
        double r75218 = r75201 ? r75211 : r75217;
        return r75218;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.3
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) < 1.0000000084552092

    1. Initial program 59.4

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

    2. Taylor expanded around 0 0.3

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

    if 1.0000000084552092 < (+ 1.0 x)

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.8

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

    5. Applied add-sqr-sqrt0.8

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

    6. Applied log-prod0.8

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

    8. Applied sqrt-unprod0.3

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

    9. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.0000000084552092:\\ \;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \log \left(\sqrt{1 + x}\right)\right) \cdot \log \left(1 + x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))