Average Error: 53.5 → 40.4
Time: 8.1s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.330733025502329 \cdot 10^{+154}:\\ \;\;\;\;\log \left(\frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq -0.005692362598187211:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{elif}\;x \leq 1.352819344293148 \cdot 10^{+154}:\\ \;\;\;\;\log \left(\frac{-1}{\frac{-1}{x + \sqrt{x \cdot x + 1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.330733025502329 \cdot 10^{+154}:\\
\;\;\;\;\log \left(\frac{-1}{x}\right)\\

\mathbf{elif}\;x \leq -0.005692362598187211:\\
\;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\

\mathbf{elif}\;x \leq 1.352819344293148 \cdot 10^{+154}:\\
\;\;\;\;\log \left(\frac{-1}{\frac{-1}{x + \sqrt{x \cdot x + 1}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log x\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.330733025502329e+154)
   (log (/ -1.0 x))
   (if (<= x -0.005692362598187211)
     (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0)))))
     (if (<= x 1.352819344293148e+154)
       (log (/ -1.0 (/ -1.0 (+ x (sqrt (+ (* x x) 1.0))))))
       (log x)))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.330733025502329e+154) {
		tmp = log(-1.0 / x);
	} else if (x <= -0.005692362598187211) {
		tmp = log(-1.0 / (x - sqrt((x * x) + 1.0)));
	} else if (x <= 1.352819344293148e+154) {
		tmp = log(-1.0 / (-1.0 / (x + sqrt((x * x) + 1.0))));
	} else {
		tmp = log(x);
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.5
Target45.6
Herbie40.4
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -1.33073302550232894e154

    1. Initial program 64.0

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied flip-+_binary64_173264.0

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{x - \sqrt{x \cdot x + 1}}\right)}\]

    4. Simplified64.0

      \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \sqrt{x \cdot x + 1}}\right)\]

    5. Taylor expanded around 0 42.9

      \[\leadsto \log \left(\frac{-1}{x - \color{blue}{0}}\right)\]

    if -1.33073302550232894e154 < x < -0.005692362598187211

    1. Initial program 61.1

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied flip-+_binary64_173260.6

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{x - \sqrt{x \cdot x + 1}}\right)}\]

    4. Simplified0.1

      \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \sqrt{x \cdot x + 1}}\right)\]

    if -0.005692362598187211 < x < 1.35281934429314794e154

    1. Initial program 47.7

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied flip-+_binary64_173259.5

      \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{x - \sqrt{x \cdot x + 1}}\right)}\]

    4. Simplified59.5

      \[\leadsto \log \left(\frac{\color{blue}{-1}}{x - \sqrt{x \cdot x + 1}}\right)\]
    5. Using strategy rm

    6. Applied flip--_binary64_173359.5

      \[\leadsto \log \left(\frac{-1}{\color{blue}{\frac{x \cdot x - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{x + \sqrt{x \cdot x + 1}}}}\right)\]

    7. Simplified47.8

      \[\leadsto \log \left(\frac{-1}{\frac{\color{blue}{-1}}{x + \sqrt{x \cdot x + 1}}}\right)\]

    if 1.35281934429314794e154 < x

    1. Initial program 64.0

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

    2. Taylor expanded around 0 42.9

      \[\leadsto \log \left(x + \color{blue}{0}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification40.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.330733025502329 \cdot 10^{+154}:\\ \;\;\;\;\log \left(\frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq -0.005692362598187211:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{elif}\;x \leq 1.352819344293148 \cdot 10^{+154}:\\ \;\;\;\;\log \left(\frac{-1}{\frac{-1}{x + \sqrt{x \cdot x + 1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array}\]

Reproduce

herbie shell --seed 2020270 
(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)))))