Average Error: 40.6 → 0.0
Time: 5.7s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
↓
\[\sqrt{\left(2 + e^{x}\right) - 1} \]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\left(2 + e^{x}\right) - 1}

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 40.6

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]

  3. Applied expm1-log1p-u_binary640.7

    \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{x}\right)\right)}} \]

  4. Applied expm1-udef_binary640.7

    \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(1 + e^{x}\right)} - 1}} \]

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2021274 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))