Average Error: 52.0 → 0.2
Time: 49.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0593607199305546:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9482022957657377:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0593607199305546:\\
\;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.9482022957657377:\\
\;\;\;\;{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(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\

\end{array}
double f(double x) {
        double r37193048 = x;
        double r37193049 = r37193048 * r37193048;
        double r37193050 = 1.0;
        double r37193051 = r37193049 + r37193050;
        double r37193052 = sqrt(r37193051);
        double r37193053 = r37193048 + r37193052;
        double r37193054 = log(r37193053);
        return r37193054;
}
↓
double f(double x) {
        double r37193055 = x;
        double r37193056 = -1.0593607199305546;
        bool r37193057 = r37193055 <= r37193056;
        double r37193058 = 0.125;
        double r37193059 = r37193058 / r37193055;
        double r37193060 = r37193059 / r37193055;
        double r37193061 = r37193060 / r37193055;
        double r37193062 = -0.5;
        double r37193063 = r37193062 / r37193055;
        double r37193064 = r37193061 + r37193063;
        double r37193065 = 0.0625;
        double r37193066 = 5.0;
        double r37193067 = pow(r37193055, r37193066);
        double r37193068 = r37193065 / r37193067;
        double r37193069 = r37193064 - r37193068;
        double r37193070 = log(r37193069);
        double r37193071 = 0.9482022957657377;
        bool r37193072 = r37193055 <= r37193071;
        double r37193073 = 0.075;
        double r37193074 = r37193067 * r37193073;
        double r37193075 = r37193055 * r37193055;
        double r37193076 = -0.16666666666666666;
        double r37193077 = r37193075 * r37193076;
        double r37193078 = r37193055 * r37193077;
        double r37193079 = r37193055 + r37193078;
        double r37193080 = r37193074 + r37193079;
        double r37193081 = 0.5;
        double r37193082 = r37193081 / r37193055;
        double r37193083 = -0.125;
        double r37193084 = r37193055 * r37193075;
        double r37193085 = r37193083 / r37193084;
        double r37193086 = r37193055 + r37193085;
        double r37193087 = r37193082 + r37193086;
        double r37193088 = r37193055 + r37193087;
        double r37193089 = log(r37193088);
        double r37193090 = r37193072 ? r37193080 : r37193089;
        double r37193091 = r37193057 ? r37193070 : r37193090;
        return r37193091;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.0
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 < -1.0593607199305546

    1. Initial program 61.6

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    if -1.0593607199305546 < x < 0.9482022957657377

    1. Initial program 58.4

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

    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

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

    if 0.9482022957657377 < x

    1. Initial program 30.5

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0593607199305546:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9482022957657377:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(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)))))