Average Error: 52.5 → 0.2
Time: 5.4s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0640612636644509:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.000671786823788058:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\\ \end{array} \]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0640612636644509:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 0.000671786823788058:\\
\;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\\


\end{array}

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.0640612636644509)
   (log (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x)))
   (if (<= x 0.000671786823788058)
     (fma (pow x 3.0) -0.16666666666666666 x)
     (log (+ x (expm1 (log1p (hypot 1.0 x))))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.0640612636644509) {
		tmp = log((0.125 / pow(x, 3.0)) + (-0.5 / x));
	} else if (x <= 0.000671786823788058) {
		tmp = fma(pow(x, 3.0), -0.16666666666666666, x);
	} else {
		tmp = log(x + expm1(log1p(hypot(1.0, x))));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original52.5
Target44.8
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 < -1.0640612636644509

    1. Initial program 61.5

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

    2. Simplified60.7

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

    3. Taylor expanded in x around -inf 0.4

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

    4. Simplified0.4

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

    if -1.0640612636644509 < x < 6.71786823788058014e-4

    1. Initial program 59.0

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

    2. Simplified59.0

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

    3. Taylor expanded in x around 0 0.2

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

    4. Simplified0.2

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

    if 6.71786823788058014e-4 < x

    1. Initial program 30.1

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

    2. Simplified0.1

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

    3. Applied expm1-log1p-u_binary640.1

      \[\leadsto \log \left(x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right)\right)\right)}\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0640612636644509:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.000671786823788058:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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)))))