Average Error: 53.4 → 0.2
Time: 6.9s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1354845962013764:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq 1.0602072132354632:\\ \;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - \left(0.044642857142857144 \cdot {x}^{7} + {x}^{3} \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.1354845962013764:\\
\;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{-1}{x}\right)\\

\mathbf{elif}\;x \leq 1.0602072132354632:\\
\;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - \left(0.044642857142857144 \cdot {x}^{7} + {x}^{3} \cdot 0.16666666666666666\right)\\

\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 -1.1354845962013764)
   (log (+ (* 0.125 (/ 1.0 (pow x 3.0))) (* 0.5 (/ -1.0 x))))
   (if (<= x 1.0602072132354632)
     (-
      (+ x (* 0.075 (pow x 5.0)))
      (+
       (* 0.044642857142857144 (pow x 7.0))
       (* (pow x 3.0) 0.16666666666666666)))
     (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 <= -1.1354845962013764) {
		tmp = log((0.125 * (1.0 / pow(x, 3.0))) + (0.5 * (-1.0 / x)));
	} else if (x <= 1.0602072132354632) {
		tmp = (x + (0.075 * pow(x, 5.0))) - ((0.044642857142857144 * pow(x, 7.0)) + (pow(x, 3.0) * 0.16666666666666666));
	} 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

Original53.4
Target45.8
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.13548459620137643

    1. Initial program 63.0

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

    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 -1.13548459620137643 < x < 1.0602072132354632

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.2

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

    if 1.0602072132354632 < x

    1. Initial program 32.6

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

    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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1354845962013764:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq 1.0602072132354632:\\ \;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - \left(0.044642857142857144 \cdot {x}^{7} + {x}^{3} \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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)))))