Average Error: 52.4 → 0.2
Time: 15.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0468290506388152:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{16}}{{x}^{5}}\right) - \frac{\frac{1}{2}}{x}\right)\\ \mathbf{elif}\;x \le 0.9618552791025287:\\ \;\;\;\;\left(x - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \frac{\frac{1}{2}}{x}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0468290506388152:\\
\;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{16}}{{x}^{5}}\right) - \frac{\frac{1}{2}}{x}\right)\\

\mathbf{elif}\;x \le 0.9618552791025287:\\
\;\;\;\;\left(x - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\

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

\end{array}
double f(double x) {
        double r5734577 = x;
        double r5734578 = r5734577 * r5734577;
        double r5734579 = 1.0;
        double r5734580 = r5734578 + r5734579;
        double r5734581 = sqrt(r5734580);
        double r5734582 = r5734577 + r5734581;
        double r5734583 = log(r5734582);
        return r5734583;
}
↓
double f(double x) {
        double r5734584 = x;
        double r5734585 = -1.0468290506388152;
        bool r5734586 = r5734584 <= r5734585;
        double r5734587 = 0.125;
        double r5734588 = r5734584 * r5734584;
        double r5734589 = r5734587 / r5734588;
        double r5734590 = r5734589 / r5734584;
        double r5734591 = 0.0625;
        double r5734592 = 5.0;
        double r5734593 = pow(r5734584, r5734592);
        double r5734594 = r5734591 / r5734593;
        double r5734595 = r5734590 - r5734594;
        double r5734596 = 0.5;
        double r5734597 = r5734596 / r5734584;
        double r5734598 = r5734595 - r5734597;
        double r5734599 = log(r5734598);
        double r5734600 = 0.9618552791025287;
        bool r5734601 = r5734584 <= r5734600;
        double r5734602 = r5734588 * r5734584;
        double r5734603 = 0.16666666666666666;
        double r5734604 = r5734602 * r5734603;
        double r5734605 = r5734584 - r5734604;
        double r5734606 = 0.075;
        double r5734607 = r5734593 * r5734606;
        double r5734608 = r5734605 + r5734607;
        double r5734609 = -0.125;
        double r5734610 = r5734609 / r5734602;
        double r5734611 = r5734610 + r5734597;
        double r5734612 = r5734611 + r5734584;
        double r5734613 = r5734584 + r5734612;
        double r5734614 = log(r5734613);
        double r5734615 = r5734601 ? r5734608 : r5734614;
        double r5734616 = r5734586 ? r5734599 : r5734615;
        return r5734616;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.4
Target44.3
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 < -1.0468290506388152

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -1.0468290506388152 < x < 0.9618552791025287

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.9618552791025287 < x

    1. Initial program 30.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0468290506388152:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{16}}{{x}^{5}}\right) - \frac{\frac{1}{2}}{x}\right)\\ \mathbf{elif}\;x \le 0.9618552791025287:\\ \;\;\;\;\left(x - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \frac{\frac{1}{2}}{x}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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)))))