Average Error: 53.4 → 11.1
Time: 7.9s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3376046051696085 \cdot 10^{+154}:\\ \;\;\;\;\log \left(\frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq -7.433249191736301 \cdot 10^{-06}:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{elif}\;x \leq 1.119329551643299 \cdot 10^{-05}:\\ \;\;\;\;x + \log 1\\ \mathbf{elif}\;x \leq 1.3218907971973426 \cdot 10^{+154}:\\ \;\;\;\;\log \left(x + \sqrt{\sqrt{x \cdot x + 1}} \cdot \sqrt{\sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.3376046051696085 \cdot 10^{+154}:\\
\;\;\;\;\log \left(\frac{-1}{x}\right)\\

\mathbf{elif}\;x \leq -7.433249191736301 \cdot 10^{-06}:\\
\;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\

\mathbf{elif}\;x \leq 1.119329551643299 \cdot 10^{-05}:\\
\;\;\;\;x + \log 1\\

\mathbf{elif}\;x \leq 1.3218907971973426 \cdot 10^{+154}:\\
\;\;\;\;\log \left(x + \sqrt{\sqrt{x \cdot x + 1}} \cdot \sqrt{\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 -1.3376046051696085e+154)
   (log (/ -1.0 x))
   (if (<= x -7.433249191736301e-06)
     (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0)))))
     (if (<= x 1.119329551643299e-05)
       (+ x (log 1.0))
       (if (<= x 1.3218907971973426e+154)
         (log
          (+
           x
           (* (sqrt (sqrt (+ (* x x) 1.0))) (sqrt (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 <= -1.3376046051696085e+154) {
		tmp = log(-1.0 / x);
	} else if (x <= -7.433249191736301e-06) {
		tmp = log(-1.0 / (x - sqrt((x * x) + 1.0)));
	} else if (x <= 1.119329551643299e-05) {
		tmp = x + log(1.0);
	} else if (x <= 1.3218907971973426e+154) {
		tmp = log(x + (sqrt(sqrt((x * x) + 1.0)) * sqrt(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

Original53.4
Target45.4
Herbie11.1
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if x < -1.3376046051696085e154

    1. Initial program 64.0

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

    3. Applied flip-+_binary6464.0

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

    4. Simplified64.0

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

    5. Taylor expanded around inf 42.9

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

    if -1.3376046051696085e154 < x < -7.43324919173630083e-6

    1. Initial program 60.3

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

    3. Applied flip-+_binary6459.8

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

    4. Simplified0.2

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

    if -7.43324919173630083e-6 < x < 1.11932955164329908e-5

    1. Initial program 59.6

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\log 1 + x}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{x + \log 1}\]

    if 1.11932955164329908e-5 < x < 1.3218907971973426e154

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt_binary640.3

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

    if 1.3218907971973426e154 < x

    1. Initial program 64.0

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

    2. Taylor expanded around inf 42.9

      \[\leadsto \log \left(x + \color{blue}{0}\right)\]
  3. Recombined 5 regimes into one program.

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3376046051696085 \cdot 10^{+154}:\\ \;\;\;\;\log \left(\frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq -7.433249191736301 \cdot 10^{-06}:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{elif}\;x \leq 1.119329551643299 \cdot 10^{-05}:\\ \;\;\;\;x + \log 1\\ \mathbf{elif}\;x \leq 1.3218907971973426 \cdot 10^{+154}:\\ \;\;\;\;\log \left(x + \sqrt{\sqrt{x \cdot x + 1}} \cdot \sqrt{\sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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)))))