Average Error: 53.0 → 0.3
Time: 17.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.013083326107210080380127692478708922863:\\ \;\;\;\;\log \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8904362184089251730512160065700300037861:\\ \;\;\;\;\left(\frac{-1}{6} \cdot \left(\frac{x}{\sqrt{1}} \cdot \frac{x \cdot x}{1}\right) + \frac{x}{\sqrt{1}}\right) + \log \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} - \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - x\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.013083326107210080380127692478708922863:\\
\;\;\;\;\log \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.8904362184089251730512160065700300037861:\\
\;\;\;\;\left(\frac{-1}{6} \cdot \left(\frac{x}{\sqrt{1}} \cdot \frac{x \cdot x}{1}\right) + \frac{x}{\sqrt{1}}\right) + \log \left(\sqrt{1}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\frac{0.5}{x} - \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - x\right)\right) + x\right)\\

\end{array}
double f(double x) {
        double r8595042 = x;
        double r8595043 = r8595042 * r8595042;
        double r8595044 = 1.0;
        double r8595045 = r8595043 + r8595044;
        double r8595046 = sqrt(r8595045);
        double r8595047 = r8595042 + r8595046;
        double r8595048 = log(r8595047);
        return r8595048;
}
↓
double f(double x) {
        double r8595049 = x;
        double r8595050 = -1.01308332610721;
        bool r8595051 = r8595049 <= r8595050;
        double r8595052 = 0.125;
        double r8595053 = r8595049 * r8595049;
        double r8595054 = r8595053 * r8595049;
        double r8595055 = r8595052 / r8595054;
        double r8595056 = 0.5;
        double r8595057 = r8595056 / r8595049;
        double r8595058 = 0.0625;
        double r8595059 = 5.0;
        double r8595060 = pow(r8595049, r8595059);
        double r8595061 = r8595058 / r8595060;
        double r8595062 = r8595057 + r8595061;
        double r8595063 = r8595055 - r8595062;
        double r8595064 = log(r8595063);
        double r8595065 = 0.8904362184089252;
        bool r8595066 = r8595049 <= r8595065;
        double r8595067 = -0.16666666666666666;
        double r8595068 = 1.0;
        double r8595069 = sqrt(r8595068);
        double r8595070 = r8595049 / r8595069;
        double r8595071 = r8595053 / r8595068;
        double r8595072 = r8595070 * r8595071;
        double r8595073 = r8595067 * r8595072;
        double r8595074 = r8595073 + r8595070;
        double r8595075 = log(r8595069);
        double r8595076 = r8595074 + r8595075;
        double r8595077 = r8595055 - r8595049;
        double r8595078 = r8595057 - r8595077;
        double r8595079 = r8595078 + r8595049;
        double r8595080 = log(r8595079);
        double r8595081 = r8595066 ? r8595076 : r8595080;
        double r8595082 = r8595051 ? r8595064 : r8595081;
        return r8595082;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.01308332610721

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{x \cdot \left(x \cdot x\right)} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)}\]

    if -1.01308332610721 < x < 0.8904362184089252

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8904362184089252 < x

    1. Initial program 31.9

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

    2. Taylor expanded around inf 0.2

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right)\]

    3. Simplified0.2

      \[\leadsto \log \left(x + \color{blue}{\left(\frac{0.5}{x} - \left(\frac{0.125}{x \cdot \left(x \cdot x\right)} - x\right)\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.013083326107210080380127692478708922863:\\ \;\;\;\;\log \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8904362184089251730512160065700300037861:\\ \;\;\;\;\left(\frac{-1}{6} \cdot \left(\frac{x}{\sqrt{1}} \cdot \frac{x \cdot x}{1}\right) + \frac{x}{\sqrt{1}}\right) + \log \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} - \left(\frac{0.125}{\left(x \cdot x\right) \cdot x} - x\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))