Hyperbolic arcsine

Average Error: 52.9 → 0.2

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0657959477224437:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9534316566480352:\\ \;\;\;\;x + \left({x}^{5} \cdot 0.075 - {x}^{3} \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.0657959477224437)
   (log (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0)))))
   (if (<= x 0.9534316566480352)
     (+ x (- (* (pow x 5.0) 0.075) (* (pow x 3.0) 0.16666666666666666)))
     (log (+ x (- (+ x (/ 0.5 x)) (/ 0.125 (pow x 3.0))))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -1.0657959477224437) {
		tmp = log((0.125 / pow(x, 3.0)) - ((0.5 / x) + (0.0625 / pow(x, 5.0))));
	} else if (x <= 0.9534316566480352) {
		tmp = x + ((pow(x, 5.0) * 0.075) - (pow(x, 3.0) * 0.16666666666666666));
	} else {
		tmp = log(x + ((x + (0.5 / x)) - (0.125 / pow(x, 3.0))));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original	52.9
Target	45.0
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.0657959477224437

   1. Initial program 62.9

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

   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 -1.0657959477224437 < x < 0.9534316566480352

   1. Initial program 58.6

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

   2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + 0.075 \cdot {x}^{5}\right) - 0.16666666666666666 \cdot {x}^{3}}\]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\left(x + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666}\]
   4. Using strategy rm

   5. Applied associate--l+_binary64_3766 0.2

      \[\leadsto \color{blue}{x + \left(0.075 \cdot {x}^{5} - {x}^{3} \cdot 0.16666666666666666\right)}\]

   if 0.9534316566480352 < x

   1. Initial program 31.5

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

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0657959477224437:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9534316566480352:\\ \;\;\;\;x + \left({x}^{5} \cdot 0.075 - {x}^{3} \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(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)))))