Average Error: 53.3 → 0.2
Time: 6.2s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0350961389186162:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9662551447614324:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + {\left(\frac{x}{\sqrt{1}}\right)}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x \cdot x} + \left(\log 2 + \left(\log x - \frac{0.09375}{{x}^{4}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0350961389186162:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \leq 0.9662551447614324:\\
\;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + {\left(\frac{x}{\sqrt{1}}\right)}^{3} \cdot -0.16666666666666666\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.25}{x \cdot x} + \left(\log 2 + \left(\log x - \frac{0.09375}{{x}^{4}}\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.0350961389186162)
   (log (- (/ 0.125 (pow x 3.0)) (+ (/ 0.5 x) (/ 0.0625 (pow x 5.0)))))
   (if (<= x 0.9662551447614324)
     (+
      (log (sqrt 1.0))
      (+ (/ x (sqrt 1.0)) (* (pow (/ x (sqrt 1.0)) 3.0) -0.16666666666666666)))
     (+ (/ 0.25 (* x x)) (+ (log 2.0) (- (log x) (/ 0.09375 (pow x 4.0))))))))
double code(double x) {
	return ((double) log(((double) (x + ((double) sqrt(((double) (((double) (x * x)) + 1.0))))))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -1.0350961389186162)) {
		tmp = ((double) log(((double) ((0.125 / ((double) pow(x, 3.0))) - ((double) ((0.5 / x) + (0.0625 / ((double) pow(x, 5.0)))))))));
	} else {
		double tmp_1;
		if ((x <= 0.9662551447614324)) {
			tmp_1 = ((double) (((double) log(((double) sqrt(1.0)))) + ((double) ((x / ((double) sqrt(1.0))) + ((double) (((double) pow((x / ((double) sqrt(1.0))), 3.0)) * -0.16666666666666666))))));
		} else {
			tmp_1 = ((double) ((0.25 / ((double) (x * x))) + ((double) (((double) log(2.0)) + ((double) (((double) log(x)) - (0.09375 / ((double) pow(x, 4.0)))))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.3
Target45.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.03509613891861618

    1. Initial program Error: 62.9 bits

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

    2. Taylor expanded around -inf Error: 0.2 bits

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]

    3. SimplifiedError: 0.2 bits

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)}\]

    if -1.03509613891861618 < x < 0.966255144761432372

    1. Initial program Error: 58.9 bits

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

    2. Taylor expanded around 0 Error: 0.2 bits

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - 0.16666666666666666 \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]

    3. SimplifiedError: 0.2 bits

      \[\leadsto \color{blue}{\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + {\left(\frac{x}{\sqrt{1}}\right)}^{3} \cdot -0.16666666666666666\right)}\]

    if 0.966255144761432372 < x

    1. Initial program Error: 31.9 bits

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

    2. Taylor expanded around inf Error: 0.3 bits

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{1}{{x}^{2}} + \log 2\right) - \left(\log \left(\frac{1}{x}\right) + 0.09375 \cdot \frac{1}{{x}^{4}}\right)}\]

    3. SimplifiedError: 0.3 bits

      \[\leadsto \color{blue}{\frac{0.25}{x \cdot x} + \left(\log 2 + \left(\log x - \frac{0.09375}{{x}^{4}}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplificationError: 0.2 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0350961389186162:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 0.9662551447614324:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{x}{\sqrt{1}} + {\left(\frac{x}{\sqrt{1}}\right)}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x \cdot x} + \left(\log 2 + \left(\log x - \frac{0.09375}{{x}^{4}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))))