Average Error: 41.4 → 41.1
Time: 7.0s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{\frac{1}{1 + \sqrt{e^{-x}}}}{1 - \sqrt{e^{-x}}}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{\frac{1}{1 + \sqrt{e^{-x}}}}{1 - \sqrt{e^{-x}}}
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (/ (/ 1.0 (+ 1.0 (sqrt (exp (- x))))) (- 1.0 (sqrt (exp (- x))))))
double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}
↓
double code(double x) {
	return (1.0 / (1.0 + sqrt(exp(-x)))) / (1.0 - sqrt(exp(-x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.4
Target41.0
Herbie41.1
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.4

    \[\frac{e^{x}}{e^{x} - 1}\]
  2. Using strategy rm

  3. Applied clear-num_binary6441.4

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

  4. Simplified41.0

    \[\leadsto \frac{1}{\color{blue}{1 - e^{-x}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt_binary6441.1

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

  7. Applied *-un-lft-identity_binary6441.1

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

  8. Applied difference-of-squares_binary6441.1

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

  9. Applied associate-/r*_binary6441.1

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

  10. Final simplification41.1

    \[\leadsto \frac{\frac{1}{1 + \sqrt{e^{-x}}}}{1 - \sqrt{e^{-x}}}\]

Reproduce

herbie shell --seed 2020268 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))