Hyperbolic arcsine

Average Error: 53.3 → 0.1

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0280831420371905:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.0012124443990565803:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right) + \left(\log \left(\sqrt{\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}}\right) + \log \left(\sqrt{\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}}\right)\right)\\ \end{array}\]

C

double code(double x) {
	return ((double) log(((double) (x + ((double) sqrt(((double) (((double) (x * x)) + 1.0))))))));
}

↓

double code(double x) {
	double VAR;
	if ((x <= -1.0280831420371905)) {
		VAR = ((double) log(((double) (((double) (0.125 / ((double) pow(x, 3.0)))) - ((double) (((double) (0.5 / x)) - ((double) (((double) -(0.0625)) / ((double) pow(x, 5.0))))))))));
	} else {
		double VAR_1;
		if ((x <= 0.0012124443990565803)) {
			VAR_1 = ((double) (((double) (((double) log(((double) sqrt(1.0)))) + ((double) (x / ((double) sqrt(1.0)))))) - ((double) (0.16666666666666666 * ((double) (((double) pow(x, 3.0)) / ((double) pow(((double) sqrt(1.0)), 3.0))))))));
		} else {
			VAR_1 = ((double) (((double) log(((double) sqrt(((double) (x + ((double) hypot(x, ((double) sqrt(1.0)))))))))) + ((double) (((double) log(((double) sqrt(((double) sqrt(((double) (x + ((double) hypot(x, ((double) sqrt(1.0)))))))))))) + ((double) log(((double) sqrt(((double) sqrt(((double) (x + ((double) hypot(x, ((double) sqrt(1.0))))))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	53.3
Target	45.3
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if x < -1.0280831420371905

   1. Initial program 63.3

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

   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. Simplified 0.1

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

   if -1.0280831420371905 < x < 0.0012124443990565803

   1. Initial program 58.7

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

   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 0.0012124443990565803 < x

   1. Initial program 32.2

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

   3. Applied add-sqr-sqrt 32.2

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

   4. Applied hypot-def 0.1

      \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(x, \sqrt{1}\right)}\right)\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 0.1

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

   7. Applied log-prod 0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right)}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 0.1

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

   10. Applied sqrt-prod 0.1

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

   11. Applied log-prod 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0280831420371905:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.0012124443990565803:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right) + \left(\log \left(\sqrt{\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}}\right) + \log \left(\sqrt{\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +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)))))