Hyperbolic arcsine

Average Error: 52.8 → 0.2

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.00943390406811484:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.0010611615811147047:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)} \cdot \sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -1.0094339040681148)) {
		VAR = log(((0.125 / pow(x, 3.0)) - ((0.5 / x) - (-0.0625 / pow(x, 5.0)))));
	} else {
		double VAR_1;
		if ((x <= 0.0010611615811147047)) {
			VAR_1 = ((log(sqrt(1.0)) + (x / sqrt(1.0))) - (0.16666666666666666 * (pow(x, 3.0) / pow(sqrt(1.0), 3.0))));
		} else {
			VAR_1 = log((sqrt((x + hypot(x, sqrt(1.0)))) * sqrt((x + hypot(x, sqrt(1.0))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	52.8
Target	45.4
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 < -1.0094339040681148

   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 -1.0094339040681148 < x < 0.0010611615811147047

   1. Initial program 58.9

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

   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 0.0010611615811147047 < x

   1. Initial program 31.1

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

   3. Applied add-sqr-sqrt 31.1

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

   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

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.00943390406811484:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.0010611615811147047:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)} \cdot \sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +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)))))