Average Error: 52.7 → 0.2
Time: 8.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.028237826169690505295761795423459261656:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.8969712032463295070527919961023144423962:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.028237826169690505295761795423459261656:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\

\mathbf{elif}\;x \le 0.8969712032463295070527919961023144423962:\\
\;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}\\

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

\end{array}
double f(double x) {
        double r162764 = x;
        double r162765 = r162764 * r162764;
        double r162766 = 1.0;
        double r162767 = r162765 + r162766;
        double r162768 = sqrt(r162767);
        double r162769 = r162764 + r162768;
        double r162770 = log(r162769);
        return r162770;
}
↓
double f(double x) {
        double r162771 = x;
        double r162772 = -1.0282378261696905;
        bool r162773 = r162771 <= r162772;
        double r162774 = 0.125;
        double r162775 = 3.0;
        double r162776 = pow(r162771, r162775);
        double r162777 = r162774 / r162776;
        double r162778 = 0.0625;
        double r162779 = 5.0;
        double r162780 = pow(r162771, r162779);
        double r162781 = r162778 / r162780;
        double r162782 = r162777 - r162781;
        double r162783 = 0.5;
        double r162784 = r162783 / r162771;
        double r162785 = r162782 - r162784;
        double r162786 = log(r162785);
        double r162787 = 0.8969712032463295;
        bool r162788 = r162771 <= r162787;
        double r162789 = 1.0;
        double r162790 = sqrt(r162789);
        double r162791 = log(r162790);
        double r162792 = -0.16666666666666666;
        double r162793 = r162771 * r162771;
        double r162794 = r162793 / r162789;
        double r162795 = r162792 * r162794;
        double r162796 = 1.0;
        double r162797 = r162795 + r162796;
        double r162798 = r162771 / r162790;
        double r162799 = r162797 * r162798;
        double r162800 = r162791 + r162799;
        double r162801 = r162771 - r162777;
        double r162802 = r162784 + r162801;
        double r162803 = r162771 + r162802;
        double r162804 = log(r162803);
        double r162805 = r162788 ? r162800 : r162804;
        double r162806 = r162773 ? r162786 : r162805;
        return r162806;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.7
Target45.0
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 < -1.0282378261696905

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -1.0282378261696905 < x < 0.8969712032463295

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8969712032463295 < x

    1. Initial program 30.9

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.028237826169690505295761795423459261656:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.8969712032463295070527919961023144423962:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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)))))