Average Error: 53.5 → 0.2
Time: 12.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0059580948780908:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90050708137317836:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(-0.125, \frac{1}{{x}^{3}}, \mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0059580948780908:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.90050708137317836:\\
\;\;\;\;\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{fma}\left(-0.125, \frac{1}{{x}^{3}}, \mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\right)\\

\end{array}
double f(double x) {
        double r172719 = x;
        double r172720 = r172719 * r172719;
        double r172721 = 1.0;
        double r172722 = r172720 + r172721;
        double r172723 = sqrt(r172722);
        double r172724 = r172719 + r172723;
        double r172725 = log(r172724);
        return r172725;
}
↓
double f(double x) {
        double r172726 = x;
        double r172727 = -1.0059580948780908;
        bool r172728 = r172726 <= r172727;
        double r172729 = 0.125;
        double r172730 = 3.0;
        double r172731 = pow(r172726, r172730);
        double r172732 = r172729 / r172731;
        double r172733 = 0.5;
        double r172734 = r172733 / r172726;
        double r172735 = 0.0625;
        double r172736 = -r172735;
        double r172737 = 5.0;
        double r172738 = pow(r172726, r172737);
        double r172739 = r172736 / r172738;
        double r172740 = r172734 - r172739;
        double r172741 = r172732 - r172740;
        double r172742 = log(r172741);
        double r172743 = 0.9005070813731784;
        bool r172744 = r172726 <= r172743;
        double r172745 = 1.0;
        double r172746 = sqrt(r172745);
        double r172747 = log(r172746);
        double r172748 = r172726 / r172746;
        double r172749 = r172747 + r172748;
        double r172750 = 0.16666666666666666;
        double r172751 = pow(r172746, r172730);
        double r172752 = r172731 / r172751;
        double r172753 = r172750 * r172752;
        double r172754 = r172749 - r172753;
        double r172755 = -r172729;
        double r172756 = 1.0;
        double r172757 = r172756 / r172731;
        double r172758 = 2.0;
        double r172759 = fma(r172726, r172758, r172734);
        double r172760 = fma(r172755, r172757, r172759);
        double r172761 = log(r172760);
        double r172762 = r172744 ? r172754 : r172761;
        double r172763 = r172728 ? r172742 : r172762;
        return r172763;
}

Error

Bits error versus x

Target

Original53.5
Target45.8
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 < -1.0059580948780908

    1. Initial program 63.0

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

    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 -1.0059580948780908 < x < 0.9005070813731784

    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 0.9005070813731784 < x

    1. Initial program 34.0

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(-0.125, \frac{1}{{x}^{3}}, \mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0059580948780908:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.90050708137317836:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(-0.125, \frac{1}{{x}^{3}}, \mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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)))))