Average Error: 52.9 → 0.1
Time: 6.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.01893041341512425:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, 1 \cdot 1, \left(-\frac{1}{2}\right) \cdot \frac{1}{x} - \frac{{1}^{3}}{\frac{{x}^{5}}{\frac{1}{16}}}\right)\right)\\ \mathbf{elif}\;x \le 9.8022759481380083 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.01893041341512425:\\
\;\;\;\;\log \left(\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, 1 \cdot 1, \left(-\frac{1}{2}\right) \cdot \frac{1}{x} - \frac{{1}^{3}}{\frac{{x}^{5}}{\frac{1}{16}}}\right)\right)\\

\mathbf{elif}\;x \le 9.8022759481380083 \cdot 10^{-4}:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

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

\end{array}
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 <= -1.0189304134151242)) {
		VAR = ((double) log(((double) fma(((double) (0.125 / ((double) pow(x, 3.0)))), ((double) (1.0 * 1.0)), ((double) (((double) (((double) -(0.5)) * ((double) (1.0 / x)))) - ((double) (((double) pow(1.0, 3.0)) / ((double) (((double) pow(x, 5.0)) / 0.0625))))))))));
	} else {
		double VAR_1;
		if ((x <= 0.0009802275948138008)) {
			VAR_1 = ((double) (((double) (((double) log(((double) sqrt(1.0)))) + ((double) (x / ((double) sqrt(1.0)))))) - ((double) (0.16666666666666666 * ((double) (((double) pow(x, 3.0)) / ((double) pow(((double) sqrt(1.0)), 3.0))))))));
		} else {
			VAR_1 = ((double) log(((double) (((double) hypot(x, ((double) sqrt(1.0)))) + x))));
		}
		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

Original52.9
Target45.3
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0189304134151242

    1. Initial program 62.9

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied add-log-exp62.9

      \[\leadsto \color{blue}{\log \left(e^{\log \left(x + \sqrt{x \cdot x + 1}\right)}\right)}\]

    4. Simplified62.9

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

    5. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{{\left(\sqrt{1}\right)}^{4}}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{{\left(\sqrt{1}\right)}^{6}}{{x}^{5}} + \frac{1}{2} \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{x}\right)\right)}\]

    6. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, 1 \cdot 1, \left(-\frac{1}{2}\right) \cdot \frac{1}{x} - \frac{{1}^{3}}{\frac{{x}^{5}}{\frac{1}{16}}}\right)\right)}\]

    if -1.0189304134151242 < x < 0.0009802275948138008

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]

    if 0.0009802275948138008 < x

    1. Initial program 31.1

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied add-log-exp31.1

      \[\leadsto \color{blue}{\log \left(e^{\log \left(x + \sqrt{x \cdot x + 1}\right)}\right)}\]

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.01893041341512425:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, 1 \cdot 1, \left(-\frac{1}{2}\right) \cdot \frac{1}{x} - \frac{{1}^{3}}{\frac{{x}^{5}}{\frac{1}{16}}}\right)\right)\\ \mathbf{elif}\;x \le 9.8022759481380083 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))