Hyperbolic arcsine

Average Error: 52.9 → 0.1

Time: 6.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.997323780249505298:\\ \;\;\;\;\log \left(1 \cdot \left(\frac{\left(1 \cdot 1\right) \cdot \frac{1}{8}}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{{\left(\sqrt{1}\right)}^{6}}{{x}^{5}} - \left(-\frac{1}{2}\right) \cdot \frac{1}{x}\right)\right)\right)\\ \mathbf{elif}\;x \le 0.001113141778355475:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 \cdot \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if ((x <= -0.9973237802495053)) {
		temp = log((1.0 * ((((1.0 * 1.0) * 0.125) / pow(x, 3.0)) - ((0.0625 * (pow(sqrt(1.0), 6.0) / pow(x, 5.0))) - (-0.5 * (1.0 / x))))));
	} else {
		double temp_1;
		if ((x <= 0.0011131417783554753)) {
			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((1.0 * (hypot(x, sqrt(1.0)) + x)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Target

Original	52.9
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 < -0.9973237802495053

   1. Initial program 63.1

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

   3. Applied *-un-lft-identity 63.1

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

   4. Applied *-un-lft-identity 63.1

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

   5. Applied distribute-lft-out 63.1

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

   6. Simplified 63.1

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

   7. Taylor expanded around -inf 0.1

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

   8. Simplified 0.1

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

   if -0.9973237802495053 < x < 0.0011131417783554753

   1. Initial program 58.8

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

   2. Taylor expanded around 0 0.2

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

   if 0.0011131417783554753 < x

   1. Initial program 30.9

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

   3. Applied *-un-lft-identity 30.9

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

   4. Applied *-un-lft-identity 30.9

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

   5. Applied distribute-lft-out 30.9

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

   6. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.997323780249505298:\\ \;\;\;\;\log \left(1 \cdot \left(\frac{\left(1 \cdot 1\right) \cdot \frac{1}{8}}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{{\left(\sqrt{1}\right)}^{6}}{{x}^{5}} - \left(-\frac{1}{2}\right) \cdot \frac{1}{x}\right)\right)\right)\\ \mathbf{elif}\;x \le 0.001113141778355475:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 \cdot \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))))