Average Error: 41.0 → 0.1
Time: 2.8s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\sqrt[3]{\left(1 + e^{x}\right) \cdot \left(\left(1 + e^{x}\right) \cdot \left(1 + e^{x}\right)\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\sqrt[3]{\left(1 + e^{x}\right) \cdot \left(\left(1 + e^{x}\right) \cdot \left(1 + e^{x}\right)\right)}}
(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
↓
(FPCore (x)
 :precision binary64
 (sqrt (cbrt (* (+ 1.0 (exp x)) (* (+ 1.0 (exp x)) (+ 1.0 (exp x)))))))
double code(double x) {
	return sqrt((exp(2.0 * x) - 1.0) / (exp(x) - 1.0));
}
↓
double code(double x) {
	return sqrt(cbrt((1.0 + exp(x)) * ((1.0 + exp(x)) * (1.0 + exp(x)))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 41.0

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

  2. Simplified0.0

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

  4. Applied add-cbrt-cube_binary640.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2021028 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))