Average Error: 53.1 → 0.1
Time: 4.7s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0125412949064443:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.0009876037437683354:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left|x + \mathsf{hypot}\left(1, x\right)\right|\right)\\ \end{array} \]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0125412949064443:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \leq 0.0009876037437683354:\\
\;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left|x + \mathsf{hypot}\left(1, x\right)\right|\right)\\


\end{array}

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.0125412949064443)
   (log (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0)))))
   (if (<= x 0.0009876037437683354)
     (fma (pow x 3.0) -0.16666666666666666 x)
     (log (fabs (+ x (hypot 1.0 x)))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.0125412949064443) {
		tmp = log((0.125 / pow(x, 3.0)) - ((0.5 / x) + (0.0625 / pow(x, 5.0))));
	} else if (x <= 0.0009876037437683354) {
		tmp = fma(pow(x, 3.0), -0.16666666666666666, x);
	} else {
		tmp = log(fabs(x + hypot(1.0, x)));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original53.1
Target45.7
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0125412949064443

    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 0.2

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

    4. 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 -1.0125412949064443 < x < 9.87603743768335424e-4

    1. Initial program 58.9

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

    2. Simplified58.9

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

    3. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{x - 0.16666666666666666 \cdot {x}^{3}} \]

    4. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \]

    if 9.87603743768335424e-4 < x

    1. Initial program 31.9

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

    2. Simplified0.0

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

    3. Applied add-sqr-sqrt_binary640.1

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

    4. Applied add-sqr-sqrt_binary640.1

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

    5. Applied rem-sqrt-square_binary640.1

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

    6. Applied add-sqr-sqrt_binary640.1

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

    7. Applied rem-sqrt-square_binary640.1

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

    8. Applied mul-fabs_binary640.1

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

    9. Simplified0.0

      \[\leadsto \log \left(\left|\color{blue}{x + \mathsf{hypot}\left(1, x\right)}\right|\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0125412949064443:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.0009876037437683354:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left|x + \mathsf{hypot}\left(1, x\right)\right|\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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)))))