Average Error: 53.2 → 0.1
Time: 13.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02274118080473863656720823200885206461:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 5.593634739476160749535593730286109348526 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02274118080473863656720823200885206461:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 5.593634739476160749535593730286109348526 \cdot 10^{-4}:\\
\;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\

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

\end{array}
double f(double x) {
        double r106654 = x;
        double r106655 = r106654 * r106654;
        double r106656 = 1.0;
        double r106657 = r106655 + r106656;
        double r106658 = sqrt(r106657);
        double r106659 = r106654 + r106658;
        double r106660 = log(r106659);
        return r106660;
}
↓
double f(double x) {
        double r106661 = x;
        double r106662 = -1.0227411808047386;
        bool r106663 = r106661 <= r106662;
        double r106664 = 0.125;
        double r106665 = 3.0;
        double r106666 = pow(r106661, r106665);
        double r106667 = r106664 / r106666;
        double r106668 = 0.5;
        double r106669 = r106668 / r106661;
        double r106670 = 0.0625;
        double r106671 = 5.0;
        double r106672 = pow(r106661, r106671);
        double r106673 = r106670 / r106672;
        double r106674 = r106669 + r106673;
        double r106675 = r106667 - r106674;
        double r106676 = log(r106675);
        double r106677 = 0.0005593634739476161;
        bool r106678 = r106661 <= r106677;
        double r106679 = 1.0;
        double r106680 = sqrt(r106679);
        double r106681 = pow(r106680, r106665);
        double r106682 = r106666 / r106681;
        double r106683 = -0.16666666666666666;
        double r106684 = log(r106680);
        double r106685 = r106661 / r106680;
        double r106686 = r106684 + r106685;
        double r106687 = fma(r106682, r106683, r106686);
        double r106688 = hypot(r106661, r106680);
        double r106689 = r106661 + r106688;
        double r106690 = log(r106689);
        double r106691 = r106678 ? r106687 : r106690;
        double r106692 = r106663 ? r106676 : r106691;
        return r106692;
}

Error

Bits error versus x

Target

Original53.2
Target45.6
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0227411808047386

    1. Initial program 63.1

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

    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

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

    if -1.0227411808047386 < x < 0.0005593634739476161

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 0.0005593634739476161 < x

    1. Initial program 31.8

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

    3. Applied add-sqr-sqrt31.8

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

    4. Applied hypot-def0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02274118080473863656720823200885206461:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 5.593634739476160749535593730286109348526 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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)))))