Average Error: 52.8 → 0.1
Time: 6.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0153027309600835:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, 1 \cdot 1, \left(-\frac{1}{2}\right) \cdot \frac{1}{x} - \frac{{1}^{3}}{\frac{{x}^{5}}{\frac{1}{16}}}\right)\right)\\ \mathbf{elif}\;x \le 8.39849586832642852 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1} \cdot \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0153027309600835:\\
\;\;\;\;\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 8.39849586832642852 \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(x + \sqrt{1} \cdot \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\

\end{array}
double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -1.0153027309600835)) {
		VAR = log(fma((0.125 / pow(x, 3.0)), (1.0 * 1.0), ((-0.5 * (1.0 / x)) - (pow(1.0, 3.0) / (pow(x, 5.0) / 0.0625)))));
	} else {
		double VAR_1;
		if ((x <= 0.0008398495868326429)) {
			VAR_1 = ((log(sqrt(1.0)) + (x / sqrt(1.0))) - (0.16666666666666666 * (pow(x, 3.0) / pow(sqrt(1.0), 3.0))));
		} else {
			VAR_1 = log((x + (sqrt(1.0) * hypot(x, sqrt(1.0)))));
		}
		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

Original52.8
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 < -1.0153027309600835

    1. Initial program 63.0

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

    3. Applied *-un-lft-identity63.0

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

    4. Applied sqrt-prod63.0

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

    5. Simplified63.0

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

    6. Taylor expanded around -inf 0.1

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

    7. Simplified0.1

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

    if -1.0153027309600835 < x < 0.0008398495868326429

    1. Initial program 58.8

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

    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 0.0008398495868326429 < x

    1. Initial program 31.0

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

    3. Applied *-un-lft-identity31.0

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

    4. Applied sqrt-prod31.0

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

    5. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0153027309600835:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{3}}, 1 \cdot 1, \left(-\frac{1}{2}\right) \cdot \frac{1}{x} - \frac{{1}^{3}}{\frac{{x}^{5}}{\frac{1}{16}}}\right)\right)\\ \mathbf{elif}\;x \le 8.39849586832642852 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1} \cdot \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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)))))