Average Error: 41.2 → 1.0
Time: 2.7s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{1}{\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{e^{x}}}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{1}{\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{e^{x}}}
double code(double x) {
	return ((double) (((double) exp(x)) / ((double) (((double) exp(x)) - 1.0))));
}
↓
double code(double x) {
	return ((double) (1.0 / ((double) (((double) (((double) (((double) pow(x, 2.0)) * ((double) (((double) (x * 0.16666666666666666)) + 0.5)))) + x)) / ((double) exp(x))))));
}

Error

Bits error versus x

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Results

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Target

Original41.2
Target40.8
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.2

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

  2. Taylor expanded around 0 11.7

    \[\leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]

  3. Simplified1.0

    \[\leadsto \frac{e^{x}}{\color{blue}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}}\]
  4. Using strategy rm

  5. Applied clear-num1.0

    \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{e^{x}}}}\]

  6. Final simplification1.0

    \[\leadsto \frac{1}{\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{e^{x}}}\]

Reproduce

herbie shell --seed 2020123 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))