Average Error: 40.6 → 0.1
Time: 4.1s
Precision: binary64
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
↓
\[\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}} \]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}

(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
↓
(FPCore (x) :precision binary64 (sqrt (/ (expm1 (+ x x)) (expm1 x))))
double code(double x) {
	return sqrt((exp(2.0 * x) - 1.0) / (exp(x) - 1.0));
}
↓
double code(double x) {
	return sqrt(expm1(x + x) / expm1(x));
}

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. Simplified0.1

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

  3. Applied flip-+_binary6440.3

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

  4. Simplified40.7

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

  5. Simplified0.1

    \[\leadsto \sqrt{\frac{-\mathsf{expm1}\left(x + x\right)}{\color{blue}{-\mathsf{expm1}\left(x\right)}}} \]

  6. Taylor expanded in x around inf 40.6

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

  7. Simplified0.1

    \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

  8. Final simplification0.1

    \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}} \]

Reproduce

herbie shell --seed 2021340 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))