Average Error: 52.8 → 0.4
Time: 5.8s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.2687108404564065:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2887980544979383:\\ \;\;\;\;x + \left(0.075 \cdot {x}^{5} - \left({x}^{3} \cdot 0.16666666666666666 + 0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.2687108404564065:\\
\;\;\;\;\log \left(\frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 1.2887980544979383:\\
\;\;\;\;x + \left(0.075 \cdot {x}^{5} - \left({x}^{3} \cdot 0.16666666666666666 + 0.044642857142857144 \cdot {x}^{7}\right)\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.2687108404564065)
   (log (/ -0.5 x))
   (if (<= x 1.2887980544979383)
     (+
      x
      (-
       (* 0.075 (pow x 5.0))
       (+
        (* (pow x 3.0) 0.16666666666666666)
        (* 0.044642857142857144 (pow x 7.0)))))
     (log (+ x x)))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.2687108404564065) {
		tmp = log(-0.5 / x);
	} else if (x <= 1.2887980544979383) {
		tmp = x + ((0.075 * pow(x, 5.0)) - ((pow(x, 3.0) * 0.16666666666666666) + (0.044642857142857144 * pow(x, 7.0))));
	} else {
		tmp = log(x + x);
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.8
Target44.9
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.26871084045640647

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 0.7

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

    if -1.26871084045640647 < x < 1.28879805449793827

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + 0.075 \cdot {x}^{5}\right) - \left(0.044642857142857144 \cdot {x}^{7} + 0.16666666666666666 \cdot {x}^{3}\right)}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(x + 0.075 \cdot {x}^{5}\right) - \left({x}^{3} \cdot 0.16666666666666666 + 0.044642857142857144 \cdot {x}^{7}\right)}\]
    4. Using strategy rm

    5. Applied associate--l+_binary64_37660.2

      \[\leadsto \color{blue}{x + \left(0.075 \cdot {x}^{5} - \left({x}^{3} \cdot 0.16666666666666666 + 0.044642857142857144 \cdot {x}^{7}\right)\right)}\]

    if 1.28879805449793827 < x

    1. Initial program 31.6

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

    2. Taylor expanded around inf 0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2687108404564065:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2887980544979383:\\ \;\;\;\;x + \left(0.075 \cdot {x}^{5} - \left({x}^{3} \cdot 0.16666666666666666 + 0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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)))))