Hyperbolic arcsine

Average Error: 53.1 → 0.1

Time: 24.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.056937052753612:\\ \;\;\;\;\log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)}\right) + \log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)}\right)\\ \mathbf{elif}\;x \le 0.007920435518071383:\\ \;\;\;\;\mathsf{fma}\left(\frac{3}{40}, {x}^{5}, \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, x\right) + x\right)\\ \end{array}\]

C

double f(double x) {
        double r5360727 = x;
        double r5360728 = r5360727 * r5360727;
        double r5360729 = 1.0;
        double r5360730 = r5360728 + r5360729;
        double r5360731 = sqrt(r5360730);
        double r5360732 = r5360727 + r5360731;
        double r5360733 = log(r5360732);
        return r5360733;
}

↓

double f(double x) {
        double r5360734 = x;
        double r5360735 = -1.056937052753612;
        bool r5360736 = r5360734 <= r5360735;
        double r5360737 = -0.0625;
        double r5360738 = 5.0;
        double r5360739 = pow(r5360734, r5360738);
        double r5360740 = r5360737 / r5360739;
        double r5360741 = 0.5;
        double r5360742 = r5360741 / r5360734;
        double r5360743 = -0.125;
        double r5360744 = r5360734 * r5360734;
        double r5360745 = r5360734 * r5360744;
        double r5360746 = r5360743 / r5360745;
        double r5360747 = r5360742 + r5360746;
        double r5360748 = r5360740 - r5360747;
        double r5360749 = sqrt(r5360748);
        double r5360750 = log(r5360749);
        double r5360751 = r5360750 + r5360750;
        double r5360752 = 0.007920435518071383;
        bool r5360753 = r5360734 <= r5360752;
        double r5360754 = 0.075;
        double r5360755 = -0.16666666666666666;
        double r5360756 = fma(r5360755, r5360745, r5360734);
        double r5360757 = fma(r5360754, r5360739, r5360756);
        double r5360758 = 1.0;
        double r5360759 = hypot(r5360758, r5360734);
        double r5360760 = r5360759 + r5360734;
        double r5360761 = log(r5360760);
        double r5360762 = r5360753 ? r5360757 : r5360761;
        double r5360763 = r5360736 ? r5360751 : r5360762;
        return r5360763;
}

Error

Bits error versus x

Target

Original	53.1
Target	45.9
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.056937052753612

   1. Initial program 61.6

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

   2. Simplified 60.8

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\]

   3. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

   4. Simplified 0.2

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \frac{\frac{1}{2}}{x}\right)\right)}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 0.2

      \[\leadsto \log \color{blue}{\left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \frac{\frac{1}{2}}{x}\right)} \cdot \sqrt{\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \frac{\frac{1}{2}}{x}\right)}\right)}\]

   7. Applied log-prod 0.2

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \frac{\frac{1}{2}}{x}\right)}\right) + \log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \frac{\frac{1}{2}}{x}\right)}\right)}\]

   if -1.056937052753612 < x < 0.007920435518071383

   1. Initial program 58.9

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

   2. Simplified 58.9

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}\]

   3. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

   4. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3}{40}, {x}^{5}, \mathsf{fma}\left(\frac{-1}{6}, \left(x \cdot x\right) \cdot x, x\right)\right)}\]

   if 0.007920435518071383 < x

   1. Initial program 32.8

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

   2. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.056937052753612:\\ \;\;\;\;\log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)}\right) + \log \left(\sqrt{\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)}\right)\\ \mathbf{elif}\;x \le 0.007920435518071383:\\ \;\;\;\;\mathsf{fma}\left(\frac{3}{40}, {x}^{5}, \mathsf{fma}\left(\frac{-1}{6}, x \cdot \left(x \cdot x\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(1, x\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))