Hyperbolic arcsine

Average Error: 53.3 → 0.3

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.999978910091019357:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.96236522228626531:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + \log \left({\left(e^{{\left(\frac{x}{\sqrt{1}}\right)}^{3}}\right)}^{\frac{-1}{6}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x \cdot x} + \left(\log 2 + \left(\log x - \frac{0.09375}{{x}^{4}}\right)\right)\\ \end{array}\]

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 <= -0.9999789100910194)) {
		VAR = ((double) log(((double) ((0.125 / ((double) pow(x, 3.0))) - ((double) ((0.5 / x) + (0.0625 / ((double) pow(x, 5.0)))))))));
	} else {
		double VAR_1;
		if ((x <= 0.9623652222862653)) {
			VAR_1 = ((double) (((double) log(((double) sqrt(1.0)))) + ((double) ((x / ((double) sqrt(1.0))) + ((double) log(((double) pow(((double) exp(((double) pow((x / ((double) sqrt(1.0))), 3.0)))), -0.16666666666666666))))))));
		} else {
			VAR_1 = ((double) ((0.25 / ((double) (x * x))) + ((double) (((double) log(2.0)) + ((double) (((double) log(x)) - (0.09375 / ((double) pow(x, 4.0)))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	53.3
Target	45.6
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x < -0.999978910091019357

   1. Initial program 62.8

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

   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. Simplified 0.2

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

   if -0.999978910091019357 < x < 0.96236522228626531

   1. Initial program 58.5

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

   2. Taylor expanded around 0 0.3

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + {\left(\frac{x}{\sqrt{1}}\right)}^{3} \cdot \frac{-1}{6}\right)}\]
   4. Using strategy rm

   5. Applied add-log-exp 0.4

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

   6. Simplified 0.4

      \[\leadsto \log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + \log \color{blue}{\left({\left(e^{{\left(\frac{x}{\sqrt{1}}\right)}^{3}}\right)}^{\frac{-1}{6}}\right)}\right)\]

   if 0.96236522228626531 < x

   1. Initial program 32.8

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

   2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{1}{{x}^{2}} + \log 2\right) - \left(\log \left(\frac{1}{x}\right) + 0.09375 \cdot \frac{1}{{x}^{4}}\right)}\]

   3. Simplified 0.3

      \[\leadsto \color{blue}{\frac{0.25}{x \cdot x} + \left(\log 2 + \left(\log x - \frac{0.09375}{{x}^{4}}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.999978910091019357:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.96236522228626531:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + \log \left({\left(e^{{\left(\frac{x}{\sqrt{1}}\right)}^{3}}\right)}^{\frac{-1}{6}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x \cdot x} + \left(\log 2 + \left(\log x - \frac{0.09375}{{x}^{4}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))