Average Error: 53.1 → 0.2
Time: 13.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.002814715336328044159586170280817896128:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.9017023301953626113203199565759859979153:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \frac{x + \frac{{x}^{3}}{\frac{1}{\frac{-1}{6}}}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.002814715336328044159586170280817896128:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\

\mathbf{elif}\;x \le 0.9017023301953626113203199565759859979153:\\
\;\;\;\;\log \left(\sqrt{1}\right) + \frac{x + \frac{{x}^{3}}{\frac{1}{\frac{-1}{6}}}}{\sqrt{1}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)\\

\end{array}
double f(double x) {
        double r111786 = x;
        double r111787 = r111786 * r111786;
        double r111788 = 1.0;
        double r111789 = r111787 + r111788;
        double r111790 = sqrt(r111789);
        double r111791 = r111786 + r111790;
        double r111792 = log(r111791);
        return r111792;
}
↓
double f(double x) {
        double r111793 = x;
        double r111794 = -1.002814715336328;
        bool r111795 = r111793 <= r111794;
        double r111796 = 0.125;
        double r111797 = 3.0;
        double r111798 = pow(r111793, r111797);
        double r111799 = r111796 / r111798;
        double r111800 = 0.0625;
        double r111801 = 5.0;
        double r111802 = pow(r111793, r111801);
        double r111803 = r111800 / r111802;
        double r111804 = r111799 - r111803;
        double r111805 = 0.5;
        double r111806 = r111805 / r111793;
        double r111807 = r111804 - r111806;
        double r111808 = log(r111807);
        double r111809 = 0.9017023301953626;
        bool r111810 = r111793 <= r111809;
        double r111811 = 1.0;
        double r111812 = sqrt(r111811);
        double r111813 = log(r111812);
        double r111814 = -0.16666666666666666;
        double r111815 = r111811 / r111814;
        double r111816 = r111798 / r111815;
        double r111817 = r111793 + r111816;
        double r111818 = r111817 / r111812;
        double r111819 = r111813 + r111818;
        double r111820 = r111806 - r111799;
        double r111821 = 2.0;
        double r111822 = r111821 * r111793;
        double r111823 = r111820 + r111822;
        double r111824 = log(r111823);
        double r111825 = r111810 ? r111819 : r111824;
        double r111826 = r111795 ? r111808 : r111825;
        return r111826;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.1
Target45.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.002814715336328

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.002814715336328 < x < 0.9017023301953626

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    5. Applied associate-*r/0.2

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

    6. Simplified0.2

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

    if 0.9017023301953626 < x

    1. Initial program 32.2

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

    2. Taylor expanded around inf 0.1

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.002814715336328044159586170280817896128:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.9017023301953626113203199565759859979153:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \frac{x + \frac{{x}^{3}}{\frac{1}{\frac{-1}{6}}}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + 2 \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))