Average Error: 41.1 → 0.1
Time: 8.1s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
↓
\[\sqrt{\frac{1 + {\left(e^{x}\right)}^{3}}{\left(1 + {\left(e^{x}\right)}^{2}\right) - e^{x}}} \]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{1 + {\left(e^{x}\right)}^{3}}{\left(1 + {\left(e^{x}\right)}^{2}\right) - e^{x}}}

(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
↓
(FPCore (x)
 :precision binary64
 (sqrt (/ (+ 1.0 (pow (exp x) 3.0)) (- (+ 1.0 (pow (exp x) 2.0)) (exp x)))))
double code(double x) {
	return sqrt((exp(2.0 * x) - 1.0) / (exp(x) - 1.0));
}
↓
double code(double x) {
	return sqrt((1.0 + pow(exp(x), 3.0)) / ((1.0 + pow(exp(x), 2.0)) - exp(x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 41.1

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
  3. Using strategy rm

  4. Applied flip3-+_binary640.1

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

  5. Simplified0.1

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

  6. Simplified0.1

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

  7. Taylor expanded in x around inf 0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2021206 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))