Hyperbolic arcsine

Average Error: 52.8 → 0.3

Time: 4.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.060913901998677:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9555061949299484:\\ \;\;\;\;\left(x + {x}^{5} \cdot 0.075\right) - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.060913901998677)
   (log (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0)))))
   (if (<= x 0.9555061949299484)
     (- (+ 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 ((double) log(((double) (x + ((double) sqrt(((double) (((double) (x * x)) + 1.0))))))));
}

↓

double code(double x) {
	double tmp;
	if ((x <= -1.060913901998677)) {
		tmp = ((double) log(((double) ((0.125 / ((double) pow(x, 3.0))) - ((double) ((0.5 / x) + (0.0625 / ((double) pow(x, 5.0)))))))));
	} else {
		double tmp_1;
		if ((x <= 0.9555061949299484)) {
			tmp_1 = ((double) (((double) (x + ((double) (((double) pow(x, 5.0)) * 0.075)))) - ((double) (((double) pow(x, 3.0)) * 0.16666666666666666))));
		} else {
			tmp_1 = ((double) log(((double) (x + ((double) (((double) (x + (0.5 / x))) - (0.125 / ((double) pow(x, 3.0)))))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Target

Original	52.8
Target	45.7
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x < -1.06091390199867708

   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 -1.06091390199867708 < x < 0.95550619492994837

   1. Initial program 58.5

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

   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

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

   if 0.95550619492994837 < x

   1. Initial program 30.5

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

   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

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.060913901998677:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9555061949299484:\\ \;\;\;\;\left(x + {x}^{5} \cdot 0.075\right) - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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)))))