Average Error: 53.2 → 0.2
Time: 8.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.99745537887981128:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 8.366604278243518 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.99745537887981128:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 8.366604278243518 \cdot 10^{-4}:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\

\end{array}
double f(double x) {
        double r186699 = x;
        double r186700 = r186699 * r186699;
        double r186701 = 1.0;
        double r186702 = r186700 + r186701;
        double r186703 = sqrt(r186702);
        double r186704 = r186699 + r186703;
        double r186705 = log(r186704);
        return r186705;
}
↓
double f(double x) {
        double r186706 = x;
        double r186707 = -0.9974553788798113;
        bool r186708 = r186706 <= r186707;
        double r186709 = 0.125;
        double r186710 = 3.0;
        double r186711 = pow(r186706, r186710);
        double r186712 = r186709 / r186711;
        double r186713 = 0.5;
        double r186714 = r186713 / r186706;
        double r186715 = 0.0625;
        double r186716 = -r186715;
        double r186717 = 5.0;
        double r186718 = pow(r186706, r186717);
        double r186719 = r186716 / r186718;
        double r186720 = r186714 - r186719;
        double r186721 = r186712 - r186720;
        double r186722 = log(r186721);
        double r186723 = 0.0008366604278243518;
        bool r186724 = r186706 <= r186723;
        double r186725 = 1.0;
        double r186726 = sqrt(r186725);
        double r186727 = log(r186726);
        double r186728 = r186706 / r186726;
        double r186729 = r186727 + r186728;
        double r186730 = 0.16666666666666666;
        double r186731 = pow(r186726, r186710);
        double r186732 = r186711 / r186731;
        double r186733 = r186730 * r186732;
        double r186734 = r186729 - r186733;
        double r186735 = hypot(r186706, r186726);
        double r186736 = r186735 + r186706;
        double r186737 = log(r186736);
        double r186738 = r186724 ? r186734 : r186737;
        double r186739 = r186708 ? r186722 : r186738;
        return r186739;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.2
Target45.5
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 < -0.9974553788798113

    1. Initial program 62.9

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

    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

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

    if -0.9974553788798113 < x < 0.0008366604278243518

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.2

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

    if 0.0008366604278243518 < x

    1. Initial program 32.0

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied add-log-exp32.0

      \[\leadsto \color{blue}{\log \left(e^{\log \left(x + \sqrt{x \cdot x + 1}\right)}\right)}\]

    4. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.99745537887981128:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 8.366604278243518 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(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)))))