Average Error: 52.9 → 0.3
Time: 6.9s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1533316581136956:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2999681543863966:\\ \;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, x\right) - \mathsf{fma}\left(0.16666666666666666, {x}^{3}, 0.044642857142857144 \cdot {x}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.1533316581136956:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\

\mathbf{elif}\;x \leq 1.2999681543863966:\\
\;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, x\right) - \mathsf{fma}\left(0.16666666666666666, {x}^{3}, 0.044642857142857144 \cdot {x}^{7}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(x \cdot 2\right)\\


\end{array}

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.1533316581136956)
   (log (+ (/ 0.125 (pow x 3.0)) (/ -0.5 x)))
   (if (<= x 1.2999681543863966)
     (-
      (fma 0.075 (pow x 5.0) x)
      (fma
       0.16666666666666666
       (pow x 3.0)
       (* 0.044642857142857144 (pow x 7.0))))
     (log (* x 2.0)))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.1533316581136956) {
		tmp = log((0.125 / pow(x, 3.0)) + (-0.5 / x));
	} else if (x <= 1.2999681543863966) {
		tmp = fma(0.075, pow(x, 5.0), x) - fma(0.16666666666666666, pow(x, 3.0), (0.044642857142857144 * pow(x, 7.0)));
	} else {
		tmp = log(x * 2.0);
	}
	return tmp;
}

Error

Bits error versus x

Target

Original52.9
Target45.0
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 < -1.1533316581136956

    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 -1.1533316581136956 < x < 1.29996815438639657

    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 0.1

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

    4. Simplified0.1

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

    if 1.29996815438639657 < x

    1. Initial program 32.0

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

    2. Simplified0.0

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

    3. Taylor expanded in x around inf 0.5

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

    4. Simplified0.5

      \[\leadsto \log \color{blue}{\left(x \cdot 2\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1533316581136956:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2999681543863966:\\ \;\;\;\;\mathsf{fma}\left(0.075, {x}^{5}, x\right) - \mathsf{fma}\left(0.16666666666666666, {x}^{3}, 0.044642857142857144 \cdot {x}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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)))))