Average Error: 52.4 → 0.2
Time: 17.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0481361170547778:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9579516068995096:\\ \;\;\;\;\log \left(e^{\frac{3}{40} \cdot {x}^{5}}\right) + \left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} + \left(\left(x + x\right) - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0481361170547778:\\
\;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\

\mathbf{elif}\;x \le 0.9579516068995096:\\
\;\;\;\;\log \left(e^{\frac{3}{40} \cdot {x}^{5}}\right) + \left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right)\\

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

\end{array}
double f(double x) {
        double r8536054 = x;
        double r8536055 = r8536054 * r8536054;
        double r8536056 = 1.0;
        double r8536057 = r8536055 + r8536056;
        double r8536058 = sqrt(r8536057);
        double r8536059 = r8536054 + r8536058;
        double r8536060 = log(r8536059);
        return r8536060;
}
↓
double f(double x) {
        double r8536061 = x;
        double r8536062 = -1.0481361170547778;
        bool r8536063 = r8536061 <= r8536062;
        double r8536064 = -0.5;
        double r8536065 = r8536064 / r8536061;
        double r8536066 = 0.0625;
        double r8536067 = 5.0;
        double r8536068 = pow(r8536061, r8536067);
        double r8536069 = r8536066 / r8536068;
        double r8536070 = 0.125;
        double r8536071 = r8536061 * r8536061;
        double r8536072 = r8536061 * r8536071;
        double r8536073 = r8536070 / r8536072;
        double r8536074 = r8536069 - r8536073;
        double r8536075 = r8536065 - r8536074;
        double r8536076 = log(r8536075);
        double r8536077 = 0.9579516068995096;
        bool r8536078 = r8536061 <= r8536077;
        double r8536079 = 0.075;
        double r8536080 = r8536079 * r8536068;
        double r8536081 = exp(r8536080);
        double r8536082 = log(r8536081);
        double r8536083 = 0.16666666666666666;
        double r8536084 = r8536072 * r8536083;
        double r8536085 = r8536061 - r8536084;
        double r8536086 = r8536082 + r8536085;
        double r8536087 = 0.5;
        double r8536088 = r8536087 / r8536061;
        double r8536089 = r8536061 + r8536061;
        double r8536090 = r8536089 - r8536073;
        double r8536091 = r8536088 + r8536090;
        double r8536092 = log(r8536091);
        double r8536093 = r8536078 ? r8536086 : r8536092;
        double r8536094 = r8536063 ? r8536076 : r8536093;
        return r8536094;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.4
Target44.4
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 < -1.0481361170547778

    1. Initial program 61.5

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

    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

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

    if -1.0481361170547778 < x < 0.9579516068995096

    1. Initial program 58.5

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

    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

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

    5. Applied add-log-exp0.2

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

    if 0.9579516068995096 < x

    1. Initial program 30.1

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0481361170547778:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9579516068995096:\\ \;\;\;\;\log \left(e^{\frac{3}{40} \cdot {x}^{5}}\right) + \left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} + \left(\left(x + x\right) - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))))