Average Error: 41.6 → 0.2
Time: 4.0s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.397327525252979 \cdot 10^{-05}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{{\left(e^{x}\right)}^{2} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.397327525252979 \cdot 10^{-05}:\\
\;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{{\left(e^{x}\right)}^{2} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\

\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 -1.397327525252979e-05)
   (sqrt
    (*
     (/ (- (pow (exp x) 2.0) 1.0) (- (pow (exp x) 2.0) (* 1.0 1.0)))
     (+ (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 <= -1.397327525252979e-05)) {
		tmp = ((double) sqrt(((double) ((((double) (((double) pow(((double) exp(x)), 2.0)) - 1.0)) / ((double) (((double) pow(((double) exp(x)), 2.0)) - ((double) (1.0 * 1.0))))) * ((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 < -1.397327525252979e-5

    1. Initial program 0.1

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

    3. Applied flip--_binary640.1

      \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]

    4. Applied associate-/r/_binary640.1

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

    5. Simplified0.0

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

    if -1.397327525252979e-5 < x

    1. Initial program 62.0

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

    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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.397327525252979 \cdot 10^{-05}:\\ \;\;\;\;\sqrt{\frac{{\left(e^{x}\right)}^{2} - 1}{{\left(e^{x}\right)}^{2} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))