Hyperbolic arcsine

Average Error: 53.1 → 0.2

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2637896809849636:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.02445914869640051:\\ \;\;\;\;\mathsf{fma}\left({x}^{7}, -0.044642857142857144, \mathsf{fma}\left({x}^{3}, -0.16666666666666666, \mathsf{fma}\left(0.075, {x}^{5}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.2637896809849636)
   (log (/ -0.5 x))
   (if (<= x 0.02445914869640051)
     (fma
      (pow x 7.0)
      -0.044642857142857144
      (fma (pow x 3.0) -0.16666666666666666 (fma 0.075 (pow x 5.0) x)))
     (log (+ x (hypot 1.0 x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -1.2637896809849636) {
		tmp = log(-0.5 / x);
	} else if (x <= 0.02445914869640051) {
		tmp = fma(pow(x, 7.0), -0.044642857142857144, fma(pow(x, 3.0), -0.16666666666666666, fma(0.075, pow(x, 5.0), x)));
	} else {
		tmp = log(x + hypot(1.0, x));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original	53.1
Target	44.9
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 < -1.263789680984964

   1. Initial program 63.1

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

   2. Simplified 63.1

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

   3. Taylor expanded in x around -inf 0.4

      \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} \]

   if -1.263789680984964 < x < 0.0244591486964005102

   1. Initial program 58.7

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

   2. Simplified 58.7

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

   3. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{\left(0.075 \cdot {x}^{5} + x\right) - \left(0.044642857142857144 \cdot {x}^{7} + 0.16666666666666666 \cdot {x}^{3}\right)} \]

   4. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{5}, x\right) - \mathsf{fma}\left(0.16666666666666666, {x}^{3}, 0.044642857142857144 \cdot {x}^{7}\right)} \]

   5. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{\left(0.075 \cdot {x}^{5} + x\right) - \left(0.044642857142857144 \cdot {x}^{7} + 0.16666666666666666 \cdot {x}^{3}\right)} \]

   6. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{7}, -0.044642857142857144, \mathsf{fma}\left({x}^{3}, -0.16666666666666666, \mathsf{fma}\left(0.075, {x}^{5}, x\right)\right)\right)} \]

   if 0.0244591486964005102 < x

   1. Initial program 31.8

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

   2. Simplified 0.0

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2637896809849636:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.02445914869640051:\\ \;\;\;\;\mathsf{fma}\left({x}^{7}, -0.044642857142857144, \mathsf{fma}\left({x}^{3}, -0.16666666666666666, \mathsf{fma}\left(0.075, {x}^{5}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(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)))))