Average Error: 53.2 → 0.1
Time: 6.8s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0255598703715518:\\ \;\;\;\;\log \left(\left(0.0390625 \cdot \frac{1}{{x}^{7}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 1.0731388620338205:\\ \;\;\;\;x + \left({x}^{5} \cdot 0.075 - \left({x}^{3} \cdot 0.16666666666666666 + {x}^{7} \cdot 0.044642857142857144\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0255598703715518:\\
\;\;\;\;\log \left(\left(0.0390625 \cdot \frac{1}{{x}^{7}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \leq 1.0731388620338205:\\
\;\;\;\;x + \left({x}^{5} \cdot 0.075 - \left({x}^{3} \cdot 0.16666666666666666 + {x}^{7} \cdot 0.044642857142857144\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.0255598703715518)
   (log
    (-
     (+ (* 0.0390625 (/ 1.0 (pow x 7.0))) (* 0.125 (/ 1.0 (pow x 3.0))))
     (+ (* 0.5 (/ 1.0 x)) (* 0.0625 (/ 1.0 (pow x 5.0))))))
   (if (<= x 1.0731388620338205)
     (+
      x
      (-
       (* (pow x 5.0) 0.075)
       (+
        (* (pow x 3.0) 0.16666666666666666)
        (* (pow x 7.0) 0.044642857142857144))))
     (log (+ x (+ x (* 0.5 (/ 1.0 x))))))))
double code(double x) {
	return log(x + sqrt((x * x) + 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -1.0255598703715518) {
		tmp = log(((0.0390625 * (1.0 / pow(x, 7.0))) + (0.125 * (1.0 / pow(x, 3.0)))) - ((0.5 * (1.0 / x)) + (0.0625 * (1.0 / pow(x, 5.0)))));
	} else if (x <= 1.0731388620338205) {
		tmp = x + ((pow(x, 5.0) * 0.075) - ((pow(x, 3.0) * 0.16666666666666666) + (pow(x, 7.0) * 0.044642857142857144)));
	} else {
		tmp = log(x + (x + (0.5 * (1.0 / 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
Target45.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.02555987037155183

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 0.1

      \[\leadsto \log \color{blue}{\left(\left(0.0390625 \cdot \frac{1}{{x}^{7}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]

    if -1.02555987037155183 < x < 1.0731388620338205

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied associate--l+_binary64_34250.1

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

    if 1.0731388620338205 < x

    1. Initial program 32.5

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0255598703715518:\\ \;\;\;\;\log \left(\left(0.0390625 \cdot \frac{1}{{x}^{7}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 1.0731388620338205:\\ \;\;\;\;x + \left({x}^{5} \cdot 0.075 - \left({x}^{3} \cdot 0.16666666666666666 + {x}^{7} \cdot 0.044642857142857144\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021019 
(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)))))