Average Error: 52.5 → 0.2
Time: 59.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0472174035704698:\\ \;\;\;\;\log \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}}\right)\right)\\ \mathbf{elif}\;x \le 0.9484625125512365:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x + x\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0472174035704698:\\
\;\;\;\;\log \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}}\right)\right)\\

\mathbf{elif}\;x \le 0.9484625125512365:\\
\;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x + x\right) + \frac{3}{40} \cdot {x}^{5}\\

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

\end{array}
double f(double x) {
        double r20322953 = x;
        double r20322954 = r20322953 * r20322953;
        double r20322955 = 1.0;
        double r20322956 = r20322954 + r20322955;
        double r20322957 = sqrt(r20322956);
        double r20322958 = r20322953 + r20322957;
        double r20322959 = log(r20322958);
        return r20322959;
}
↓
double f(double x) {
        double r20322960 = x;
        double r20322961 = -1.0472174035704698;
        bool r20322962 = r20322960 <= r20322961;
        double r20322963 = 0.125;
        double r20322964 = r20322963 / r20322960;
        double r20322965 = r20322964 / r20322960;
        double r20322966 = r20322965 / r20322960;
        double r20322967 = -0.5;
        double r20322968 = r20322967 / r20322960;
        double r20322969 = r20322966 + r20322968;
        double r20322970 = 0.0625;
        double r20322971 = 5.0;
        double r20322972 = pow(r20322960, r20322971);
        double r20322973 = r20322970 / r20322972;
        double r20322974 = r20322969 - r20322973;
        double r20322975 = cbrt(r20322974);
        double r20322976 = r20322975 * r20322975;
        double r20322977 = r20322975 * r20322976;
        double r20322978 = log(r20322977);
        double r20322979 = 0.9484625125512365;
        bool r20322980 = r20322960 <= r20322979;
        double r20322981 = r20322960 * r20322960;
        double r20322982 = -0.16666666666666666;
        double r20322983 = r20322981 * r20322982;
        double r20322984 = r20322983 * r20322960;
        double r20322985 = r20322984 + r20322960;
        double r20322986 = 0.075;
        double r20322987 = r20322986 * r20322972;
        double r20322988 = r20322985 + r20322987;
        double r20322989 = 0.5;
        double r20322990 = r20322989 / r20322960;
        double r20322991 = -0.125;
        double r20322992 = r20322981 * r20322960;
        double r20322993 = r20322991 / r20322992;
        double r20322994 = r20322993 + r20322960;
        double r20322995 = r20322990 + r20322994;
        double r20322996 = r20322960 + r20322995;
        double r20322997 = log(r20322996);
        double r20322998 = r20322980 ? r20322988 : r20322997;
        double r20322999 = r20322962 ? r20322978 : r20322998;
        return r20322999;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.5
Target44.8
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 < -1.0472174035704698

    1. Initial program 61.7

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

    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

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

    5. Applied add-cube-cbrt0.2

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

    if -1.0472174035704698 < x < 0.9484625125512365

    1. Initial program 58.6

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

    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 0.9484625125512365 < x

    1. Initial program 31.2

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0472174035704698:\\ \;\;\;\;\log \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}}\right)\right)\\ \mathbf{elif}\;x \le 0.9484625125512365:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x + x\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(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)))))