Average Error: 52.6 → 0.2
Time: 25.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0612400279589977:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9581083600858236:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0612400279589977:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9581083600858236:\\
\;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\

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

\end{array}
double f(double x) {
        double r7157071 = x;
        double r7157072 = r7157071 * r7157071;
        double r7157073 = 1.0;
        double r7157074 = r7157072 + r7157073;
        double r7157075 = sqrt(r7157074);
        double r7157076 = r7157071 + r7157075;
        double r7157077 = log(r7157076);
        return r7157077;
}
↓
double f(double x) {
        double r7157078 = x;
        double r7157079 = -1.0612400279589977;
        bool r7157080 = r7157078 <= r7157079;
        double r7157081 = -0.0625;
        double r7157082 = r7157078 * r7157078;
        double r7157083 = r7157082 * r7157078;
        double r7157084 = r7157082 * r7157083;
        double r7157085 = r7157081 / r7157084;
        double r7157086 = 0.125;
        double r7157087 = r7157086 / r7157078;
        double r7157088 = r7157087 / r7157082;
        double r7157089 = 0.5;
        double r7157090 = r7157089 / r7157078;
        double r7157091 = r7157088 - r7157090;
        double r7157092 = r7157085 + r7157091;
        double r7157093 = log(r7157092);
        double r7157094 = 0.9581083600858236;
        bool r7157095 = r7157078 <= r7157094;
        double r7157096 = 0.075;
        double r7157097 = r7157096 * r7157084;
        double r7157098 = -0.16666666666666666;
        double r7157099 = r7157083 * r7157098;
        double r7157100 = r7157097 + r7157099;
        double r7157101 = r7157100 + r7157078;
        double r7157102 = r7157078 + r7157078;
        double r7157103 = r7157088 - r7157102;
        double r7157104 = r7157090 - r7157103;
        double r7157105 = log(r7157104);
        double r7157106 = r7157095 ? r7157101 : r7157105;
        double r7157107 = r7157080 ? r7157093 : r7157106;
        return r7157107;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.6
Target44.7
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 < -1.0612400279589977

    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

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

    if -1.0612400279589977 < x < 0.9581083600858236

    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

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

    if 0.9581083600858236 < x

    1. Initial program 31.3

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0612400279589977:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9581083600858236:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))))