Hyperbolic arcsine

Average Error: 52.6 → 0.2

Time: 18.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0788943454777058:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9628322546007364:\\ \;\;\;\;x + \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{3}{40} - \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r5374770 = x;
        double r5374771 = r5374770 * r5374770;
        double r5374772 = 1.0;
        double r5374773 = r5374771 + r5374772;
        double r5374774 = sqrt(r5374773);
        double r5374775 = r5374770 + r5374774;
        double r5374776 = log(r5374775);
        return r5374776;
}

↓

double f(double x) {
        double r5374777 = x;
        double r5374778 = -1.0788943454777058;
        bool r5374779 = r5374777 <= r5374778;
        double r5374780 = -0.0625;
        double r5374781 = r5374777 * r5374777;
        double r5374782 = r5374781 * r5374777;
        double r5374783 = r5374781 * r5374782;
        double r5374784 = r5374780 / r5374783;
        double r5374785 = 0.125;
        double r5374786 = r5374785 / r5374782;
        double r5374787 = 0.5;
        double r5374788 = r5374787 / r5374777;
        double r5374789 = r5374786 - r5374788;
        double r5374790 = r5374784 + r5374789;
        double r5374791 = log(r5374790);
        double r5374792 = 0.9628322546007364;
        bool r5374793 = r5374777 <= r5374792;
        double r5374794 = 0.075;
        double r5374795 = r5374781 * r5374794;
        double r5374796 = 0.16666666666666666;
        double r5374797 = r5374795 - r5374796;
        double r5374798 = r5374782 * r5374797;
        double r5374799 = r5374777 + r5374798;
        double r5374800 = -0.125;
        double r5374801 = r5374800 / r5374782;
        double r5374802 = r5374801 + r5374777;
        double r5374803 = r5374788 + r5374802;
        double r5374804 = r5374777 + r5374803;
        double r5374805 = log(r5374804);
        double r5374806 = r5374793 ? r5374799 : r5374805;
        double r5374807 = r5374779 ? r5374791 : r5374806;
        return r5374807;
}

Error

Bits error versus x

Target

Original	52.6
Target	44.8
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x < -1.0788943454777058

   1. Initial program 61.6

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

   2. Taylor expanded around -inf 0.2

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

   3. Simplified 0.2

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

   if -1.0788943454777058 < x < 0.9628322546007364

   1. Initial program 58.5

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

   2. Taylor expanded around 0 0.2

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

   3. Simplified 0.2

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

   4. Taylor expanded around -inf 0.2

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

   5. Simplified 0.2

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

   if 0.9628322546007364 < x

   1. Initial program 30.9

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified 0.2

      \[\leadsto \log \left(x + \color{blue}{\left(\left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right) + \frac{\frac{1}{2}}{x}\right)}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0788943454777058:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9628322546007364:\\ \;\;\;\;x + \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{3}{40} - \frac{1}{6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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)))))