Average Error: 53.2 → 0.1
Time: 8.0s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0229856676033233:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)\\ \mathbf{elif}\;x \leq 0.0010396709414883317:\\ \;\;\;\;x - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0229856676033233:\\
\;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)\\

\mathbf{elif}\;x \leq 0.0010396709414883317:\\
\;\;\;\;x - {x}^{3} \cdot 0.16666666666666666\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\


\end{array}

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.0229856676033233)
   (log
    (-
     (* 0.125 (/ 1.0 (pow x 3.0)))
     (+ (* 0.0625 (/ 1.0 (pow x 5.0))) (* 0.5 (/ 1.0 x)))))
   (if (<= x 0.0010396709414883317)
     (- x (* (pow x 3.0) 0.16666666666666666))
     (log (+ x (hypot 1.0 x))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.0229856676033233) {
		tmp = log((0.125 * (1.0 / pow(x, 3.0))) - ((0.0625 * (1.0 / pow(x, 5.0))) + (0.5 * (1.0 / x))));
	} else if (x <= 0.0010396709414883317) {
		tmp = x - (pow(x, 3.0) * 0.16666666666666666);
	} else {
		tmp = log(x + hypot(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

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

Derivation

  1. Split input into 3 regimes

  2. if x < -1.02298566760332332

    1. Initial program 62.8

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

    2. Simplified62.8

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

    3. Taylor expanded in x around -inf 0.2

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)} \]

    if -1.02298566760332332 < x < 0.0010396709414883317

    1. Initial program 58.9

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

    2. Simplified58.9

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

    3. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{x - 0.16666666666666666 \cdot {x}^{3}} \]

    if 0.0010396709414883317 < x

    1. Initial program 32.4

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

    2. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0229856676033233:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)\\ \mathbf{elif}\;x \leq 0.0010396709414883317:\\ \;\;\;\;x - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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)))))