Hyperbolic arcsine

Average Error: 53.0 → 0.1

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0296783040954165:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, 1 \cdot 1, \left(-\frac{1}{2}\right) \cdot \frac{1}{x} - \frac{{1}^{3}}{\frac{{x}^{5}}{\frac{1}{16}}}\right)\right)\\ \mathbf{elif}\;x \le 0.00106203832322045752:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if ((x <= -1.0296783040954165)) {
		temp = log(fma((0.125 / pow(x, 3.0)), (1.0 * 1.0), ((-0.5 * (1.0 / x)) - (pow(1.0, 3.0) / (pow(x, 5.0) / 0.0625)))));
	} else {
		double temp_1;
		if ((x <= 0.0010620383232204575)) {
			temp_1 = ((log(sqrt(1.0)) + (x / sqrt(1.0))) - (0.16666666666666666 * (pow(x, 3.0) / pow(sqrt(1.0), 3.0))));
		} else {
			temp_1 = log((x + hypot(x, sqrt(1.0))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Target

Original	53.0
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 < -1.0296783040954165

   1. Initial program 62.9

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

   3. Applied add-sqr-sqrt 62.9

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

   4. Applied hypot-def 62.9

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

   5. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{{\left(\sqrt{1}\right)}^{6}}{{x}^{5}} + \frac{1}{2} \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{x}\right)\right)}\]

   6. Simplified 0.2

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

   if -1.0296783040954165 < x < 0.0010620383232204575

   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 0.0010620383232204575 < x

   1. Initial program 31.8

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

   3. Applied add-sqr-sqrt 31.8

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

   4. Applied hypot-def 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0296783040954165:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, 1 \cdot 1, \left(-\frac{1}{2}\right) \cdot \frac{1}{x} - \frac{{1}^{3}}{\frac{{x}^{5}}{\frac{1}{16}}}\right)\right)\\ \mathbf{elif}\;x \le 0.00106203832322045752:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))))