Hyperbolic arcsine

Average Error: 52.9 → 0.1

Time: 20.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0432984552881022:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9442966733188264:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + \frac{\frac{1}{2}}{x}\right) + \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) + x\right)\\ \end{array}\]

C

double f(double x) {
        double r5818491 = x;
        double r5818492 = r5818491 * r5818491;
        double r5818493 = 1.0;
        double r5818494 = r5818492 + r5818493;
        double r5818495 = sqrt(r5818494);
        double r5818496 = r5818491 + r5818495;
        double r5818497 = log(r5818496);
        return r5818497;
}

↓

double f(double x) {
        double r5818498 = x;
        double r5818499 = -1.0432984552881022;
        bool r5818500 = r5818498 <= r5818499;
        double r5818501 = -0.0625;
        double r5818502 = r5818498 * r5818498;
        double r5818503 = r5818502 * r5818498;
        double r5818504 = r5818502 * r5818503;
        double r5818505 = r5818501 / r5818504;
        double r5818506 = 0.125;
        double r5818507 = r5818506 / r5818498;
        double r5818508 = r5818507 / r5818502;
        double r5818509 = 0.5;
        double r5818510 = r5818509 / r5818498;
        double r5818511 = r5818508 - r5818510;
        double r5818512 = r5818505 + r5818511;
        double r5818513 = log(r5818512);
        double r5818514 = 0.9442966733188264;
        bool r5818515 = r5818498 <= r5818514;
        double r5818516 = 0.075;
        double r5818517 = r5818516 * r5818504;
        double r5818518 = -0.16666666666666666;
        double r5818519 = r5818503 * r5818518;
        double r5818520 = r5818517 + r5818519;
        double r5818521 = r5818520 + r5818498;
        double r5818522 = r5818498 + r5818510;
        double r5818523 = -0.125;
        double r5818524 = r5818523 / r5818503;
        double r5818525 = r5818522 + r5818524;
        double r5818526 = r5818525 + r5818498;
        double r5818527 = log(r5818526);
        double r5818528 = r5818515 ? r5818521 : r5818527;
        double r5818529 = r5818500 ? r5818513 : r5818528;
        return r5818529;
}

Error

Bits error versus x

Target

Original	52.9
Target	45.4
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 < -1.0432984552881022

   1. Initial program 61.9

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

   2. Taylor expanded around -inf 0.1

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

   3. Simplified 0.1

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

   if -1.0432984552881022 < x < 0.9442966733188264

   1. Initial program 58.6

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

   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

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

   if 0.9442966733188264 < x

   1. Initial program 31.5

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

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0432984552881022:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9442966733188264:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + \frac{\frac{1}{2}}{x}\right) + \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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)))))