Average Error: 53.2 → 0.2
Time: 4.9s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.9960767900717074:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9002939444000406:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + {\left(\frac{x}{\sqrt{1}}\right)}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.9960767900717074:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \leq 0.9002939444000406:\\
\;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + {\left(\frac{x}{\sqrt{1}}\right)}^{3} \cdot -0.16666666666666666\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.9960767900717074)
   (log (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0)))))
   (if (<= x 0.9002939444000406)
     (+
      (log (sqrt 1.0))
      (+ (/ x (sqrt 1.0)) (* (pow (/ x (sqrt 1.0)) 3.0) -0.16666666666666666)))
     (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 VAR;
	if ((x <= -0.9960767900717074)) {
		VAR = ((double) log(((double) ((0.125 / ((double) pow(x, 3.0))) - ((double) ((0.5 / x) + (0.0625 / ((double) pow(x, 5.0)))))))));
	} else {
		double VAR_1;
		if ((x <= 0.9002939444000406)) {
			VAR_1 = ((double) (((double) log(((double) sqrt(1.0)))) + ((double) ((x / ((double) sqrt(1.0))) + ((double) (((double) pow((x / ((double) sqrt(1.0))), 3.0)) * -0.16666666666666666))))));
		} else {
			VAR_1 = ((double) log(((double) (x + ((double) (x + ((double) ((0.5 / x) - (0.125 / ((double) pow(x, 3.0)))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.99607679007170735

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -0.99607679007170735 < x < 0.90029394440004062

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 0.90029394440004062 < x

    1. Initial program 32.6

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9960767900717074:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9002939444000406:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + {\left(\frac{x}{\sqrt{1}}\right)}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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)))))