Average Error: 52.5 → 0.2
Time: 9.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0757153356474918:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9619268999163055:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0757153356474918:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9619268999163055:\\
\;\;\;\;\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}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\

\end{array}
double f(double x) {
        double r2508073 = x;
        double r2508074 = r2508073 * r2508073;
        double r2508075 = 1.0;
        double r2508076 = r2508074 + r2508075;
        double r2508077 = sqrt(r2508076);
        double r2508078 = r2508073 + r2508077;
        double r2508079 = log(r2508078);
        return r2508079;
}
↓
double f(double x) {
        double r2508080 = x;
        double r2508081 = -1.0757153356474918;
        bool r2508082 = r2508080 <= r2508081;
        double r2508083 = -0.0625;
        double r2508084 = 5.0;
        double r2508085 = pow(r2508080, r2508084);
        double r2508086 = r2508083 / r2508085;
        double r2508087 = 0.125;
        double r2508088 = r2508080 * r2508080;
        double r2508089 = r2508080 * r2508088;
        double r2508090 = r2508087 / r2508089;
        double r2508091 = -0.5;
        double r2508092 = r2508091 / r2508080;
        double r2508093 = r2508090 + r2508092;
        double r2508094 = r2508086 + r2508093;
        double r2508095 = log(r2508094);
        double r2508096 = 0.9619268999163055;
        bool r2508097 = r2508080 <= r2508096;
        double r2508098 = 0.075;
        double r2508099 = r2508085 * r2508098;
        double r2508100 = -0.16666666666666666;
        double r2508101 = r2508089 * r2508100;
        double r2508102 = r2508099 + r2508101;
        double r2508103 = r2508102 + r2508080;
        double r2508104 = 0.5;
        double r2508105 = r2508104 / r2508080;
        double r2508106 = r2508105 - r2508090;
        double r2508107 = r2508080 + r2508106;
        double r2508108 = r2508107 + r2508080;
        double r2508109 = log(r2508108);
        double r2508110 = r2508097 ? r2508103 : r2508109;
        double r2508111 = r2508082 ? r2508095 : r2508110;
        return r2508111;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.5
Target45.1
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 < -1.0757153356474918

    1. Initial program 61.6

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

    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 -1.0757153356474918 < x < 0.9619268999163055

    1. Initial program 58.7

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

    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

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

    5. Applied associate-+l+0.2

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

    if 0.9619268999163055 < x

    1. Initial program 31.5

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0757153356474918:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9619268999163055:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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)))))