Average Error: 41.5 → 0.0
Time: 5.3s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
↓
\[\sqrt{e^{\log \left(1 + e^{x}\right)}} \]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{e^{\log \left(1 + e^{x}\right)}}

(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
↓
(FPCore (x) :precision binary64 (sqrt (exp (log (+ 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(exp(log(1.0 + 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 41.5

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

  2. Simplified0.0

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

  3. Applied add-exp-log_binary640.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022097 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))