Average Error: 53.2 → 0.4
Time: 4.1s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.9051123451222073:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) - \frac{0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.7855249968658321:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.9051123451222073:\\
\;\;\;\;\log \left(\frac{-0.5}{x}\right) - \frac{0.25}{x \cdot x}\\

\mathbf{elif}\;x \leq 0.7855249968658321:\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.9051123451222073)
   (- (log (/ -0.5 x)) (/ 0.25 (* x x)))
   (if (<= x 0.7855249968658321) x (log (+ x (+ x (/ 0.5 x)))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.9051123451222073) {
		tmp = log(-0.5 / x) - (0.25 / (x * x));
	} else if (x <= 0.7855249968658321) {
		tmp = x;
	} else {
		tmp = log(x + (x + (0.5 / x)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.2
Target45.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.905112345122207285

    1. Initial program 62.6

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

    2. Taylor expanded around -inf 0.5

      \[\leadsto \color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}}\]

    3. Simplified0.5

      \[\leadsto \color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right) - \frac{0.25}{x \cdot x}}\]
    4. Using strategy rm

    5. Applied sum-log_binary640.4

      \[\leadsto \color{blue}{\log \left(0.5 \cdot \frac{-1}{x}\right)} - \frac{0.25}{x \cdot x}\]

    6. Simplified0.4

      \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} - \frac{0.25}{x \cdot x}\]

    if -0.905112345122207285 < x < 0.785524996865832104

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{x}\]

    if 0.785524996865832104 < x

    1. Initial program 31.9

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right)\]

    3. Simplified0.3

      \[\leadsto \log \left(x + \color{blue}{\left(x + \frac{0.5}{x}\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9051123451222073:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right) - \frac{0.25}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.7855249968658321:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))))