Average Error: 39.3 → 0.3
Time: 4.1s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.0000019383723648:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(1 + x\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt{\sqrt{1 + x}}\right)\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.0000019383723648:\\
\;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\

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

\end{array}
double f(double x) {
        double r68240 = 1.0;
        double r68241 = x;
        double r68242 = r68240 + r68241;
        double r68243 = log(r68242);
        return r68243;
}
↓
double f(double x) {
        double r68244 = 1.0;
        double r68245 = x;
        double r68246 = r68244 + r68245;
        double r68247 = 1.0000019383723648;
        bool r68248 = r68246 <= r68247;
        double r68249 = log(r68244);
        double r68250 = 0.5;
        double r68251 = 2.0;
        double r68252 = pow(r68245, r68251);
        double r68253 = pow(r68244, r68251);
        double r68254 = r68252 / r68253;
        double r68255 = r68250 * r68254;
        double r68256 = r68249 - r68255;
        double r68257 = fma(r68245, r68244, r68256);
        double r68258 = log(r68246);
        double r68259 = r68250 * r68258;
        double r68260 = sqrt(r68246);
        double r68261 = sqrt(r68260);
        double r68262 = log(r68261);
        double r68263 = r68262 + r68262;
        double r68264 = r68259 + r68263;
        double r68265 = r68248 ? r68257 : r68264;
        return r68265;
}

Error

Bits error versus x

Target

Original39.3
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) < 1.0000019383723648

    1. Initial program 59.1

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

    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

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

    if 1.0000019383723648 < (+ 1.0 x)

    1. Initial program 0.1

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

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

    6. Applied pow10.2

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

    7. Applied sqrt-pow10.2

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

    8. Applied log-pow0.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \log \left(1 + x\right)} + \log \left(\sqrt{1 + x}\right)\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt0.2

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

    11. Applied sqrt-prod0.2

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

    12. Applied log-prod0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.0000019383723648:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(1 + x\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt{\sqrt{1 + x}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +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)))