Average Error: 41.0 → 0.3
Time: 3.9s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0001300712652255136:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left({\left(e^{x}\right)}^{2} - 1\right)}^{3}}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.0001300712652255136:\\
\;\;\;\;\sqrt{\frac{\sqrt[3]{{\left({\left(e^{x}\right)}^{2} - 1\right)}^{3}}}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\

\end{array}
(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.0001300712652255136)
   (sqrt (/ (cbrt (pow (- (pow (exp x) 2.0) 1.0) 3.0)) (- (exp x) 1.0)))
   (sqrt (+ 2.0 (* x (+ 1.0 (* x 0.5)))))))
double code(double x) {
	return ((double) sqrt((((double) (((double) exp(((double) (2.0 * x)))) - 1.0)) / ((double) (((double) exp(x)) - 1.0)))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.0001300712652255136)) {
		tmp = ((double) sqrt((((double) cbrt(((double) pow(((double) (((double) pow(((double) exp(x)), 2.0)) - 1.0)), 3.0)))) / ((double) (((double) exp(x)) - 1.0)))));
	} else {
		tmp = ((double) sqrt(((double) (2.0 + ((double) (x * ((double) (1.0 + ((double) (x * 0.5))))))))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -1.30071265225513612e-4

    1. Initial program 0.0

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube_binary640.0

      \[\leadsto \sqrt{\frac{\color{blue}{\sqrt[3]{\left(\left(e^{2 \cdot x} - 1\right) \cdot \left(e^{2 \cdot x} - 1\right)\right) \cdot \left(e^{2 \cdot x} - 1\right)}}}{e^{x} - 1}}\]

    4. Simplified0.0

      \[\leadsto \sqrt{\frac{\sqrt[3]{\color{blue}{{\left({\left(e^{x}\right)}^{2} - 1\right)}^{3}}}}{e^{x} - 1}}\]

    if -1.30071265225513612e-4 < x

    1. Initial program 61.7

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

    2. Taylor expanded around 0 0.4

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]

    3. Simplified0.4

      \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(x \cdot 0.5 + 1\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0001300712652255136:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left({\left(e^{x}\right)}^{2} - 1\right)}^{3}}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))