Average Error: 52.5 → 0.3
Time: 4.6s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1457422521714031:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3153391522167552:\\ \;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.1457422521714031:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 1.3153391522167552:\\
\;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\\

\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 -1.1457422521714031)
   (log (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x)))
   (if (<= x 1.3153391522167552)
     (- (+ x (* 0.075 (pow x 5.0))) (* (pow x 3.0) 0.16666666666666666))
     (log (+ x x)))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.1457422521714031) {
		tmp = log((0.125 / pow(x, 3.0)) + (-0.5 / x));
	} else if (x <= 1.3153391522167552) {
		tmp = (x + (0.075 * pow(x, 5.0))) - (pow(x, 3.0) * 0.16666666666666666);
	} 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

Original52.5
Target44.9
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 < -1.1457422521714031

    1. Initial program 62.9

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

    2. Simplified62.9

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

    3. Taylor expanded in x around -inf 0.3

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

    4. Simplified0.3

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

    if -1.1457422521714031 < x < 1.31533915221675524

    1. Initial program 58.6

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

    2. Simplified58.6

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

    3. Taylor expanded in x around 0 0.2

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

    if 1.31533915221675524 < x

    1. Initial program 30.7

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]

    3. Applied add-log-exp_binary6463.5

      \[\leadsto \log \left(x + \color{blue}{\log \left(e^{\mathsf{hypot}\left(1, x\right)}\right)}\right) \]

    4. Applied add-log-exp_binary6463.5

      \[\leadsto \log \left(\color{blue}{\log \left(e^{x}\right)} + \log \left(e^{\mathsf{hypot}\left(1, x\right)}\right)\right) \]

    5. Applied sum-log_binary6463.5

      \[\leadsto \log \color{blue}{\log \left(e^{x} \cdot e^{\mathsf{hypot}\left(1, x\right)}\right)} \]

    6. Simplified63.5

      \[\leadsto \log \log \color{blue}{\left(e^{x + \mathsf{hypot}\left(1, x\right)}\right)} \]

    7. Taylor expanded in x around inf 0.5

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

    8. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1457422521714031:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3153391522167552:\\ \;\;\;\;\left(x + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022087 
(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)))))