Average Error: 41.0 → 0.5
Time: 5.1s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{0.5}{\sinh x} + \frac{0.5}{\sinh x} \cdot e^{x}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{0.5}{\sinh x} + \frac{0.5}{\sinh x} \cdot e^{x}
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (+ (/ 0.5 (sinh x)) (* (/ 0.5 (sinh x)) (exp x))))
double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}
↓
double code(double x) {
	return (0.5 / sinh(x)) + ((0.5 / sinh(x)) * exp(x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.0
Target40.6
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.0

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

  3. Applied flip--_binary6441.1

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

  4. Applied associate-/r/_binary6441.1

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

  5. Simplified41.0

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

  7. Applied sinh-undef_binary640.5

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

  8. Applied associate-/r*_binary640.5

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

  9. Simplified0.5

    \[\leadsto \frac{\color{blue}{0.5}}{\sinh x} \cdot \left(e^{x} + 1\right)\]
  10. Using strategy rm

  11. Applied distribute-rgt-in_binary640.5

    \[\leadsto \color{blue}{e^{x} \cdot \frac{0.5}{\sinh x} + 1 \cdot \frac{0.5}{\sinh x}}\]

  12. Simplified0.5

    \[\leadsto \color{blue}{\frac{0.5}{\sinh x} \cdot e^{x}} + 1 \cdot \frac{0.5}{\sinh x}\]

  13. Simplified0.5

    \[\leadsto \frac{0.5}{\sinh x} \cdot e^{x} + \color{blue}{\frac{0.5}{\sinh x}}\]

  14. Final simplification0.5

    \[\leadsto \frac{0.5}{\sinh x} + \frac{0.5}{\sinh x} \cdot e^{x}\]

Reproduce

herbie shell --seed 2020288 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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