Average Error: 53.0 → 0.2
Time: 17.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02122159882250018725358131632674485445:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8950674725377756324462552584009245038033:\\ \;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02122159882250018725358131632674485445:\\
\;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8950674725377756324462552584009245038033:\\
\;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\

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

\end{array}
double f(double x) {
        double r7712760 = x;
        double r7712761 = r7712760 * r7712760;
        double r7712762 = 1.0;
        double r7712763 = r7712761 + r7712762;
        double r7712764 = sqrt(r7712763);
        double r7712765 = r7712760 + r7712764;
        double r7712766 = log(r7712765);
        return r7712766;
}
↓
double f(double x) {
        double r7712767 = x;
        double r7712768 = -1.0212215988225002;
        bool r7712769 = r7712767 <= r7712768;
        double r7712770 = 0.125;
        double r7712771 = r7712770 / r7712767;
        double r7712772 = r7712767 * r7712767;
        double r7712773 = r7712771 / r7712772;
        double r7712774 = 0.0625;
        double r7712775 = 5.0;
        double r7712776 = pow(r7712767, r7712775);
        double r7712777 = r7712774 / r7712776;
        double r7712778 = 0.5;
        double r7712779 = r7712778 / r7712767;
        double r7712780 = r7712777 + r7712779;
        double r7712781 = r7712773 - r7712780;
        double r7712782 = log(r7712781);
        double r7712783 = 0.8950674725377756;
        bool r7712784 = r7712767 <= r7712783;
        double r7712785 = 1.0;
        double r7712786 = sqrt(r7712785);
        double r7712787 = r7712767 / r7712786;
        double r7712788 = log(r7712786);
        double r7712789 = 0.16666666666666666;
        double r7712790 = r7712772 * r7712767;
        double r7712791 = r7712790 / r7712785;
        double r7712792 = r7712786 / r7712791;
        double r7712793 = r7712789 / r7712792;
        double r7712794 = r7712788 - r7712793;
        double r7712795 = r7712787 + r7712794;
        double r7712796 = r7712767 - r7712773;
        double r7712797 = r7712779 + r7712796;
        double r7712798 = r7712767 + r7712797;
        double r7712799 = log(r7712798);
        double r7712800 = r7712784 ? r7712795 : r7712799;
        double r7712801 = r7712769 ? r7712782 : r7712800;
        return r7712801;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.3
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 < -1.0212215988225002

    1. Initial program 62.8

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    if -1.0212215988225002 < x < 0.8950674725377756

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8950674725377756 < x

    1. Initial program 32.1

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02122159882250018725358131632674485445:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8950674725377756324462552584009245038033:\\ \;\;\;\;\frac{x}{\sqrt{1}} + \left(\log \left(\sqrt{1}\right) - \frac{\frac{1}{6}}{\frac{\sqrt{1}}{\frac{\left(x \cdot x\right) \cdot x}{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))