Average Error: 53.2 → 0.3
Time: 4.7s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.262667212110832:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.0461861871581324:\\ \;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - \left(0.044642857142857144 \cdot {x}^{7} + 0.16666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.262667212110832:\\
\;\;\;\;\log \left(\frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 1.0461861871581324:\\
\;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - \left(0.044642857142857144 \cdot {x}^{7} + 0.16666666666666666 \cdot {x}^{3}\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.262667212110832)
   (log (/ -0.5 x))
   (if (<= x 1.0461861871581324)
     (-
      (+ x (* 0.075 (pow x 5.0)))
      (+
       (* 0.044642857142857144 (pow x 7.0))
       (* 0.16666666666666666 (pow x 3.0))))
     (log (+ x (+ x (/ 0.5 x)))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.262667212110832) {
		tmp = log(-0.5 / x);
	} else if (x <= 1.0461861871581324) {
		tmp = (x + (0.075 * pow(x, 5.0))) - ((0.044642857142857144 * pow(x, 7.0)) + (0.16666666666666666 * pow(x, 3.0)));
	} else {
		tmp = log(x + (x + (0.5 / x)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.2
Target45.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.262667212110832

    1. Initial program 62.8

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

    2. Taylor expanded around -inf 0.6

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

    if -1.262667212110832 < x < 1.04618618715813239

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.1

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

    if 1.04618618715813239 < x

    1. Initial program 32.9

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.262667212110832:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.0461861871581324:\\ \;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - \left(0.044642857142857144 \cdot {x}^{7} + 0.16666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(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)))))