Average Error: 53.3 → 0.2
Time: 4.8s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1570741760631404:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.0257505134280076:\\ \;\;\;\;\left(x - {x}^{3} \cdot 0.16666666666666666\right) + 0.075 \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.1570741760631404:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 1.0257505134280076:\\
\;\;\;\;\left(x - {x}^{3} \cdot 0.16666666666666666\right) + 0.075 \cdot {x}^{5}\\

\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 -1.1570741760631404)
   (log (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x)))
   (if (<= x 1.0257505134280076)
     (+ (- x (* (pow x 3.0) 0.16666666666666666)) (* 0.075 (pow x 5.0)))
     (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 <= -1.1570741760631404) {
		tmp = log((0.125 / pow(x, 3.0)) + (-0.5 / x));
	} else if (x <= 1.0257505134280076) {
		tmp = (x - (pow(x, 3.0) * 0.16666666666666666)) + (0.075 * pow(x, 5.0));
	} 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
Target45.6
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 < -1.15707417606314045

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.15707417606314045 < x < 1.0257505134280076

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 1.0257505134280076 < x

    1. Initial program 31.8

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1570741760631404:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.0257505134280076:\\ \;\;\;\;\left(x - {x}^{3} \cdot 0.16666666666666666\right) + 0.075 \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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)))))