Average Error: 52.7 → 0.2
Time: 31.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0816386065582386:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9440471341056946:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0816386065582386:\\
\;\;\;\;\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 0.9440471341056946:\\
\;\;\;\;\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 r6371694 = x;
        double r6371695 = r6371694 * r6371694;
        double r6371696 = 1.0;
        double r6371697 = r6371695 + r6371696;
        double r6371698 = sqrt(r6371697);
        double r6371699 = r6371694 + r6371698;
        double r6371700 = log(r6371699);
        return r6371700;
}
↓
double f(double x) {
        double r6371701 = x;
        double r6371702 = -1.0816386065582386;
        bool r6371703 = r6371701 <= r6371702;
        double r6371704 = -0.0625;
        double r6371705 = r6371701 * r6371701;
        double r6371706 = r6371705 * r6371701;
        double r6371707 = r6371705 * r6371706;
        double r6371708 = r6371704 / r6371707;
        double r6371709 = 0.125;
        double r6371710 = r6371709 / r6371701;
        double r6371711 = r6371710 / r6371705;
        double r6371712 = 0.5;
        double r6371713 = r6371712 / r6371701;
        double r6371714 = r6371711 - r6371713;
        double r6371715 = r6371708 + r6371714;
        double r6371716 = log(r6371715);
        double r6371717 = 0.9440471341056946;
        bool r6371718 = r6371701 <= r6371717;
        double r6371719 = 0.075;
        double r6371720 = r6371719 * r6371707;
        double r6371721 = -0.16666666666666666;
        double r6371722 = r6371706 * r6371721;
        double r6371723 = r6371720 + r6371722;
        double r6371724 = r6371723 + r6371701;
        double r6371725 = r6371701 + r6371701;
        double r6371726 = r6371711 - r6371725;
        double r6371727 = r6371713 - r6371726;
        double r6371728 = log(r6371727);
        double r6371729 = r6371718 ? r6371724 : r6371728;
        double r6371730 = r6371703 ? r6371716 : r6371729;
        return r6371730;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.7
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 < -1.0816386065582386

    1. Initial program 61.5

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

    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

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

    if -1.0816386065582386 < x < 0.9440471341056946

    1. Initial program 58.6

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

    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

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

    if 0.9440471341056946 < x

    1. Initial program 31.8

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0816386065582386:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9440471341056946:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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)))))