Average Error: 52.7 → 0.3
Time: 16.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0840635159626253:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9540314055762552:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x}\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0840635159626253:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\

\mathbf{elif}\;x \le 0.9540314055762552:\\
\;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + x\\

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

\end{array}
double f(double x) {
        double r8147961 = x;
        double r8147962 = r8147961 * r8147961;
        double r8147963 = 1.0;
        double r8147964 = r8147962 + r8147963;
        double r8147965 = sqrt(r8147964);
        double r8147966 = r8147961 + r8147965;
        double r8147967 = log(r8147966);
        return r8147967;
}
↓
double f(double x) {
        double r8147968 = x;
        double r8147969 = -1.0840635159626253;
        bool r8147970 = r8147968 <= r8147969;
        double r8147971 = -0.0625;
        double r8147972 = 5.0;
        double r8147973 = pow(r8147968, r8147972);
        double r8147974 = r8147971 / r8147973;
        double r8147975 = 0.5;
        double r8147976 = r8147975 / r8147968;
        double r8147977 = 0.125;
        double r8147978 = r8147968 * r8147968;
        double r8147979 = r8147968 * r8147978;
        double r8147980 = r8147977 / r8147979;
        double r8147981 = r8147976 - r8147980;
        double r8147982 = r8147974 - r8147981;
        double r8147983 = log(r8147982);
        double r8147984 = 0.9540314055762552;
        bool r8147985 = r8147968 <= r8147984;
        double r8147986 = 0.075;
        double r8147987 = r8147973 * r8147986;
        double r8147988 = -0.16666666666666666;
        double r8147989 = r8147979 * r8147988;
        double r8147990 = r8147987 + r8147989;
        double r8147991 = r8147990 + r8147968;
        double r8147992 = r8147980 - r8147976;
        double r8147993 = r8147968 - r8147992;
        double r8147994 = r8147993 + r8147968;
        double r8147995 = log(r8147994);
        double r8147996 = r8147985 ? r8147991 : r8147995;
        double r8147997 = r8147970 ? r8147983 : r8147996;
        return r8147997;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.7
Target44.8
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0840635159626253

    1. Initial program 61.8

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

    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

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

    if -1.0840635159626253 < x < 0.9540314055762552

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.9540314055762552 < x

    1. Initial program 32.3

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0840635159626253:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9540314055762552:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} - \frac{\frac{1}{2}}{x}\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))))