Average Error: 52.8 → 0.5
Time: 4.4s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.9629144065967837:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 0.7968192574558219:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.9629144065967837:\\
\;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)\\

\mathbf{elif}\;x \leq 0.7968192574558219:\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.9629144065967837)
   (log (- (* 0.125 (/ 1.0 (pow x 3.0))) (* 0.5 (/ 1.0 x))))
   (if (<= x 0.7968192574558219) x (log (+ x (+ x (* 0.5 (/ 1.0 x))))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.9629144065967837) {
		tmp = log((0.125 * (1.0 / pow(x, 3.0))) - (0.5 * (1.0 / x)));
	} else if (x <= 0.7968192574558219) {
		tmp = x;
	} else {
		tmp = log(x + (x + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.8
Target45.0
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.962914406596783734

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.3

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

    if -0.962914406596783734 < x < 0.79681925745582194

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{x}\]

    if 0.79681925745582194 < x

    1. Initial program 30.8

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9629144065967837:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 0.7968192574558219:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021020 
(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)))))