Average Error: 40.6 → 0.1
Time: 1.1min
Precision: binary64
Cost: 1472
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{\frac{1 + {\left(e^{x}\right)}^{2} \cdot {\left(\sqrt{e^{x}}\right)}^{2}}{1 + \left({\left(e^{x}\right)}^{2} - e^{x}\right)}}\]
\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{\frac{1 + {\left(e^{x}\right)}^{2} \cdot {\left(\sqrt{e^{x}}\right)}^{2}}{1 + \left({\left(e^{x}\right)}^{2} - e^{x}\right)}}
(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) 2.0) (pow (sqrt (exp x)) 2.0)))
   (+ 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), 2.0) * pow(sqrt(exp(x)), 2.0))) / (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
Alternative 1
Accuracy0.1
Cost1280
\[\sqrt{\frac{1 + e^{x} \cdot {\left(e^{x}\right)}^{2}}{1 + \left({\left(e^{x}\right)}^{2} - e^{x}\right)}}\]
Alternative 2
Accuracy0.1
Cost1088
\[\sqrt{\frac{1 + {\left(e^{x}\right)}^{3}}{1 + \left({\left(e^{x}\right)}^{2} - e^{x}\right)}}\]

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. 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. Using strategy rm

  8. Applied cube-mult_binary640.1

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

  9. Simplified0.1

    \[\leadsto \sqrt{\frac{1 + e^{x} \cdot \color{blue}{{\left(e^{x}\right)}^{2}}}{1 + \left({\left(e^{x}\right)}^{2} - e^{x}\right)}}\]
  10. Using strategy rm

  11. Applied add-sqr-sqrt_binary640.1

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

  12. Applied unpow-prod-down_binary640.1

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

  13. Applied associate-*r*_binary640.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020322 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))