Average Error: 52.5 → 0.1
Time: 1.2m
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0715992056992538:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9586135763949255:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} - x\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0715992056992538:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} - \frac{\frac{1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9586135763949255:\\
\;\;\;\;{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(\left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} - x\right)\right) + x\right)\\

\end{array}
double f(double x) {
        double r30659058 = x;
        double r30659059 = r30659058 * r30659058;
        double r30659060 = 1.0;
        double r30659061 = r30659059 + r30659060;
        double r30659062 = sqrt(r30659061);
        double r30659063 = r30659058 + r30659062;
        double r30659064 = log(r30659063);
        return r30659064;
}
↓
double f(double x) {
        double r30659065 = x;
        double r30659066 = -1.0715992056992538;
        bool r30659067 = r30659065 <= r30659066;
        double r30659068 = -0.0625;
        double r30659069 = 5.0;
        double r30659070 = pow(r30659065, r30659069);
        double r30659071 = r30659068 / r30659070;
        double r30659072 = 0.125;
        double r30659073 = r30659072 / r30659065;
        double r30659074 = r30659073 / r30659065;
        double r30659075 = r30659074 / r30659065;
        double r30659076 = 0.5;
        double r30659077 = r30659076 / r30659065;
        double r30659078 = r30659075 - r30659077;
        double r30659079 = r30659071 + r30659078;
        double r30659080 = log(r30659079);
        double r30659081 = 0.9586135763949255;
        bool r30659082 = r30659065 <= r30659081;
        double r30659083 = 0.075;
        double r30659084 = r30659070 * r30659083;
        double r30659085 = r30659065 * r30659065;
        double r30659086 = -0.16666666666666666;
        double r30659087 = r30659085 * r30659086;
        double r30659088 = r30659065 * r30659087;
        double r30659089 = r30659065 + r30659088;
        double r30659090 = r30659084 + r30659089;
        double r30659091 = r30659065 * r30659085;
        double r30659092 = r30659072 / r30659091;
        double r30659093 = r30659092 - r30659065;
        double r30659094 = r30659077 - r30659093;
        double r30659095 = r30659094 + r30659065;
        double r30659096 = log(r30659095);
        double r30659097 = r30659082 ? r30659090 : r30659096;
        double r30659098 = r30659067 ? r30659080 : r30659097;
        return r30659098;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.5
Target44.8
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0715992056992538

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -1.0715992056992538 < x < 0.9586135763949255

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 0.9586135763949255 < x

    1. Initial program 30.8

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0715992056992538:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9586135763949255:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} - x\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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)))))