Average Error: 40.0 → 0.9
Time: 15.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}
double f(double x) {
        double r4605620 = x;
        double r4605621 = exp(r4605620);
        double r4605622 = 1.0;
        double r4605623 = r4605621 - r4605622;
        double r4605624 = r4605621 / r4605623;
        return r4605624;
}
↓
double f(double x) {
        double r4605625 = x;
        double r4605626 = exp(r4605625);
        double r4605627 = 0.5;
        double r4605628 = 0.16666666666666666;
        double r4605629 = r4605628 * r4605625;
        double r4605630 = r4605627 + r4605629;
        double r4605631 = r4605625 * r4605625;
        double r4605632 = r4605630 * r4605631;
        double r4605633 = r4605625 + r4605632;
        double r4605634 = r4605626 / r4605633;
        return r4605634;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target39.7
Herbie0.9
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.0

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

  2. Taylor expanded around 0 11.5

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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