Average Error: 53.4 → 0.2
Time: 4.1s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1513240953628694:\\ \;\;\;\;\log \left(\sqrt[3]{\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}} \cdot \left(\sqrt[3]{\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}}\right)\right)\\ \mathbf{elif}\;x \leq 1.0399647248488046:\\ \;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.1513240953628694:\\
\;\;\;\;\log \left(\sqrt[3]{\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}} \cdot \left(\sqrt[3]{\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}}\right)\right)\\

\mathbf{elif}\;x \leq 1.0399647248488046:\\
\;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.1513240953628694)
   (log
    (*
     (cbrt (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x)))
     (*
      (cbrt (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x)))
      (cbrt (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x))))))
   (if (<= x 1.0399647248488046)
     (- (+ x (* 0.075 (pow x 5.0))) (* (pow x 3.0) 0.16666666666666666))
     (log (+ x (+ x (* 0.5 (/ 1.0 x))))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.1513240953628694) {
		tmp = log(cbrt((0.125 / pow(x, 3.0)) + (-0.5 / x)) * (cbrt((0.125 / pow(x, 3.0)) + (-0.5 / x)) * cbrt((0.125 / pow(x, 3.0)) + (-0.5 / x))));
	} else if (x <= 1.0399647248488046) {
		tmp = (x + (0.075 * pow(x, 5.0))) - (pow(x, 3.0) * 0.16666666666666666);
	} else {
		tmp = log(x + (x + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.4
Target45.2
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 < -1.1513240953628694

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    5. Applied add-cube-cbrt_binary640.3

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

    if -1.1513240953628694 < x < 1.03996472484880464

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.2

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

    if 1.03996472484880464 < x

    1. Initial program 32.9

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1513240953628694:\\ \;\;\;\;\log \left(\sqrt[3]{\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}} \cdot \left(\sqrt[3]{\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}}\right)\right)\\ \mathbf{elif}\;x \leq 1.0399647248488046:\\ \;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021113 
(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)))))