Average Error: 52.9 → 0.3
Time: 4.6s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.071505362580263:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 1.0126904852337015:\\ \;\;\;\;\left(x + {x}^{5} \cdot 0.075\right) - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.25}{x \cdot x} + \log 2\right) - \left(\frac{0.09375}{{x}^{4}} - \log x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.071505362580263:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \leq 1.0126904852337015:\\
\;\;\;\;\left(x + {x}^{5} \cdot 0.075\right) - {x}^{3} \cdot 0.16666666666666666\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{0.25}{x \cdot x} + \log 2\right) - \left(\frac{0.09375}{{x}^{4}} - \log x\right)\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.071505362580263)
   (log (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0)))))
   (if (<= x 1.0126904852337015)
     (- (+ x (* (pow x 5.0) 0.075)) (* (pow x 3.0) 0.16666666666666666))
     (- (+ (/ 0.25 (* x x)) (log 2.0)) (- (/ 0.09375 (pow x 4.0)) (log x))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.071505362580263) {
		tmp = log((0.125 / pow(x, 3.0)) - ((0.5 / x) + (0.0625 / pow(x, 5.0))));
	} else if (x <= 1.0126904852337015) {
		tmp = (x + (pow(x, 5.0) * 0.075)) - (pow(x, 3.0) * 0.16666666666666666);
	} else {
		tmp = ((0.25 / (x * x)) + log(2.0)) - ((0.09375 / pow(x, 4.0)) - log(x));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target45.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.07150536258026308

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.07150536258026308 < x < 1.01269048523370153

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 1.01269048523370153 < x

    1. Initial program 32.6

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

    2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{1}{{x}^{2}} + \log 2\right) - \left(\log \left(\frac{1}{x}\right) + 0.09375 \cdot \frac{1}{{x}^{4}}\right)}\]

    3. Simplified0.4

      \[\leadsto \color{blue}{\left(\frac{0.25}{x \cdot x} + \log 2\right) - \left(\frac{0.09375}{{x}^{4}} - \log x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.071505362580263:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 1.0126904852337015:\\ \;\;\;\;\left(x + {x}^{5} \cdot 0.075\right) - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.25}{x \cdot x} + \log 2\right) - \left(\frac{0.09375}{{x}^{4}} - \log x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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)))))