Average Error: 52.8 → 0.1
Time: 4.9s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1375845794427093:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.02314138012114505:\\ \;\;\;\;\mathsf{fma}\left({x}^{7}, -0.044642857142857144, \mathsf{fma}\left({x}^{3}, -0.16666666666666666, \mathsf{fma}\left(0.075, {x}^{5}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.1375845794427093:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 0.02314138012114505:\\
\;\;\;\;\mathsf{fma}\left({x}^{7}, -0.044642857142857144, \mathsf{fma}\left({x}^{3}, -0.16666666666666666, \mathsf{fma}\left(0.075, {x}^{5}, x\right)\right)\right)\\

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


\end{array}

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.1375845794427093)
   (log (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x)))
   (if (<= x 0.02314138012114505)
     (fma
      (pow x 7.0)
      -0.044642857142857144
      (fma (pow x 3.0) -0.16666666666666666 (fma 0.075 (pow x 5.0) x)))
     (log (+ 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 <= -1.1375845794427093) {
		tmp = log((0.125 / pow(x, 3.0)) + (-0.5 / x));
	} else if (x <= 0.02314138012114505) {
		tmp = fma(pow(x, 7.0), -0.044642857142857144, fma(pow(x, 3.0), -0.16666666666666666, fma(0.075, pow(x, 5.0), x)));
	} else {
		tmp = log(x + hypot(1.0, x));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original52.8
Target45.5
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 < -1.137584579442709

    1. Initial program 63.1

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

    2. Simplified63.1

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

    3. Taylor expanded in x around -inf 0.2

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

    4. Simplified0.2

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

    if -1.137584579442709 < x < 0.023141380121145048

    1. Initial program 58.7

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

    2. Simplified58.7

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

    3. Taylor expanded in x around 0 0.0

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

    4. Simplified0.0

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

    5. Taylor expanded in x around 0 0.0

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

    6. Simplified0.0

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

    if 0.023141380121145048 < x

    1. Initial program 31.1

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

    2. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1375845794427093:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.02314138012114505:\\ \;\;\;\;\mathsf{fma}\left({x}^{7}, -0.044642857142857144, \mathsf{fma}\left({x}^{3}, -0.16666666666666666, \mathsf{fma}\left(0.075, {x}^{5}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(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)))))