Average Error: 53.3 → 0.2
Time: 4.4s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0089138002281648:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right) + \frac{0.0390625}{{x}^{7}}\right)\\ \mathbf{elif}\;x \leq 1.0440251375487621:\\ \;\;\;\;\left(x - {x}^{3} \cdot 0.16666666666666666\right) + {x}^{5} \cdot 0.075\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0089138002281648:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right) + \frac{0.0390625}{{x}^{7}}\right)\\

\mathbf{elif}\;x \leq 1.0440251375487621:\\
\;\;\;\;\left(x - {x}^{3} \cdot 0.16666666666666666\right) + {x}^{5} \cdot 0.075\\

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


\end{array}

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.0089138002281648)
   (log
    (+
     (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0))))
     (/ 0.0390625 (pow x 7.0))))
   (if (<= x 1.0440251375487621)
     (+ (- x (* (pow x 3.0) 0.16666666666666666)) (* (pow x 5.0) 0.075))
     (log (+ x (+ x (/ 0.5 x)))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.0089138002281648) {
		tmp = log(((0.125 / pow(x, 3.0)) - ((0.5 / x) + (0.0625 / pow(x, 5.0)))) + (0.0390625 / pow(x, 7.0)));
	} else if (x <= 1.0440251375487621) {
		tmp = (x - (pow(x, 3.0) * 0.16666666666666666)) + (pow(x, 5.0) * 0.075);
	} else {
		tmp = log(x + (x + (0.5 / x)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.3
Target45.5
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 < -1.0089138002281648

    1. Initial program 62.6

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\left(0.0390625 \cdot \frac{1}{{x}^{7}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)} \]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right) + \frac{0.0390625}{{x}^{7}}\right)} \]

    if -1.0089138002281648 < x < 1.0440251375487621

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(0.075 \cdot {x}^{5} + x\right) - 0.16666666666666666 \cdot {x}^{3}} \]

    3. Simplified0.1

      \[\leadsto \color{blue}{\left(x - {x}^{3} \cdot 0.16666666666666666\right) + {x}^{5} \cdot 0.075} \]

    if 1.0440251375487621 < x

    1. Initial program 32.8

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0089138002281648:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right) + \frac{0.0390625}{{x}^{7}}\right)\\ \mathbf{elif}\;x \leq 1.0440251375487621:\\ \;\;\;\;\left(x - {x}^{3} \cdot 0.16666666666666666\right) + {x}^{5} \cdot 0.075\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021197 
(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)))))