Average Error: 41.0 → 1.0
Time: 3.4s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 1\right)}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 1\right)}
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (/ (exp x) (* x (+ (* x (+ 0.5 (* x 0.16666666666666666))) 1.0))))
double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}
↓
double code(double x) {
	return exp(x) / (x * ((x * (0.5 + (x * 0.16666666666666666))) + 1.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.0
Target40.6
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.0

    \[\frac{e^{x}}{e^{x} - 1}\]

  2. Taylor expanded around 0 11.5

    \[\leadsto \frac{e^{x}}{\color{blue}{0.5 \cdot {x}^{2} + \left(0.16666666666666666 \cdot {x}^{3} + x\right)}}\]

  3. Simplified1.0

    \[\leadsto \frac{e^{x}}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 1\right)}}\]

  4. Final simplification1.0

    \[\leadsto \frac{e^{x}}{x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 1\right)}\]

Reproduce

herbie shell --seed 2020231 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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