Average Error: 52.8 → 0.3
Time: 4.3s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.010460937713024:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.887192876086426:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - 0.16666666666666666 \cdot {\left(\frac{x}{\sqrt{1}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.010460937713024:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \leq 0.887192876086426:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - 0.16666666666666666 \cdot {\left(\frac{x}{\sqrt{1}}\right)}^{3}\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.010460937713024)
   (log (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0)))))
   (if (<= x 0.887192876086426)
     (-
      (+ (log (sqrt 1.0)) (/ x (sqrt 1.0)))
      (* 0.16666666666666666 (pow (/ x (sqrt 1.0)) 3.0)))
     (log (+ x (- (+ x (/ 0.5 x)) (/ 0.125 (pow x 3.0))))))))
double code(double x) {
	return ((double) log(((double) (x + ((double) sqrt(((double) (((double) (x * x)) + 1.0))))))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -1.010460937713024)) {
		tmp = ((double) log(((double) ((0.125 / ((double) pow(x, 3.0))) - ((double) ((0.5 / x) + (0.0625 / ((double) pow(x, 5.0)))))))));
	} else {
		double tmp_1;
		if ((x <= 0.887192876086426)) {
			tmp_1 = ((double) (((double) (((double) log(((double) sqrt(1.0)))) + (x / ((double) sqrt(1.0))))) - ((double) (0.16666666666666666 * ((double) pow((x / ((double) sqrt(1.0))), 3.0))))));
		} else {
			tmp_1 = ((double) log(((double) (x + ((double) (((double) (x + (0.5 / x))) - (0.125 / ((double) pow(x, 3.0)))))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.8
Target45.7
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 < -1.01046093771302403

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.01046093771302403 < x < 0.88719287608642605

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - 0.16666666666666666 \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]

    3. Simplified0.3

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - 0.16666666666666666 \cdot {\left(\frac{x}{\sqrt{1}}\right)}^{3}}\]

    if 0.88719287608642605 < x

    1. Initial program 30.5

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.010460937713024:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.887192876086426:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - 0.16666666666666666 \cdot {\left(\frac{x}{\sqrt{1}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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)))))