Hyperbolic arcsine

Average Error: 53.5 → 0.2

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.02614982763272589:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.89282765018932997:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} - \frac{1}{6} \cdot {\left(\frac{x}{\sqrt{1}}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

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 <= -1.0261498276327259)) {
		VAR = ((double) log(((double) (((double) (0.125 / ((double) pow(x, 3.0)))) - ((double) (((double) (0.5 / x)) + ((double) (0.0625 / ((double) pow(x, 5.0))))))))));
	} else {
		double VAR_1;
		if ((x <= 0.89282765018933)) {
			VAR_1 = ((double) (((double) log(((double) sqrt(1.0)))) + ((double) (((double) (x / ((double) sqrt(1.0)))) - ((double) (0.16666666666666666 * ((double) pow(((double) (x / ((double) sqrt(1.0)))), 3.0))))))));
		} else {
			VAR_1 = ((double) log(((double) (x + ((double) (x + ((double) (((double) (0.5 / x)) - ((double) (0.125 / ((double) pow(x, 3.0))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	53.5
Target	45.8
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x < -1.02614982763272589

   1. Initial program 63.1

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

   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 -1.02614982763272589 < x < 0.89282765018932997

   1. Initial program 58.8

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

   2. Taylor expanded around 0 0.2

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

   3. Simplified 0.2

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

   if 0.89282765018932997 < x

   1. Initial program 32.5

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

   2. Taylor expanded around inf 0.3

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right)\]

   3. Simplified 0.3

      \[\leadsto \log \left(x + \color{blue}{\left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02614982763272589:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.89282765018932997:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} - \frac{1}{6} \cdot {\left(\frac{x}{\sqrt{1}}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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)))))