Average Error: 52.8 → 0.2
Time: 40.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0695644432902263:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9660867967165637:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0695644432902263:\\
\;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.9660867967165637:\\
\;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\

\end{array}
double f(double x) {
        double r37738962 = x;
        double r37738963 = r37738962 * r37738962;
        double r37738964 = 1.0;
        double r37738965 = r37738963 + r37738964;
        double r37738966 = sqrt(r37738965);
        double r37738967 = r37738962 + r37738966;
        double r37738968 = log(r37738967);
        return r37738968;
}
↓
double f(double x) {
        double r37738969 = x;
        double r37738970 = -1.0695644432902263;
        bool r37738971 = r37738969 <= r37738970;
        double r37738972 = 0.125;
        double r37738973 = r37738972 / r37738969;
        double r37738974 = r37738973 / r37738969;
        double r37738975 = r37738974 / r37738969;
        double r37738976 = -0.5;
        double r37738977 = r37738976 / r37738969;
        double r37738978 = r37738975 + r37738977;
        double r37738979 = 0.0625;
        double r37738980 = 5.0;
        double r37738981 = pow(r37738969, r37738980);
        double r37738982 = r37738979 / r37738981;
        double r37738983 = r37738978 - r37738982;
        double r37738984 = log(r37738983);
        double r37738985 = 0.9660867967165637;
        bool r37738986 = r37738969 <= r37738985;
        double r37738987 = 0.075;
        double r37738988 = r37738981 * r37738987;
        double r37738989 = r37738969 * r37738969;
        double r37738990 = -0.16666666666666666;
        double r37738991 = r37738989 * r37738990;
        double r37738992 = r37738969 * r37738991;
        double r37738993 = r37738969 + r37738992;
        double r37738994 = r37738988 + r37738993;
        double r37738995 = 0.5;
        double r37738996 = r37738995 / r37738969;
        double r37738997 = -0.125;
        double r37738998 = r37738969 * r37738989;
        double r37738999 = r37738997 / r37738998;
        double r37739000 = r37738969 + r37738999;
        double r37739001 = r37738996 + r37739000;
        double r37739002 = r37738969 + r37739001;
        double r37739003 = log(r37739002);
        double r37739004 = r37738986 ? r37738994 : r37739003;
        double r37739005 = r37738971 ? r37738984 : r37739004;
        return r37739005;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.8
Target45.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0695644432902263

    1. Initial program 61.9

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\left(\frac{\frac{-1}{2}}{x} + \frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)}\]

    if -1.0695644432902263 < x < 0.9660867967165637

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x + x\right) + {x}^{5} \cdot \frac{3}{40}}\]

    if 0.9660867967165637 < x

    1. Initial program 32.1

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

    2. Taylor expanded around inf 0.2

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)}\right)\]

    3. Simplified0.2

      \[\leadsto \log \left(x + \color{blue}{\left(\left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right) + \frac{\frac{1}{2}}{x}\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0695644432902263:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9660867967165637:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))