Average Error: 41.4 → 0.8
Time: 11.6s
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 r60648 = x;
        double r60649 = exp(r60648);
        double r60650 = 1.0;
        double r60651 = r60649 - r60650;
        double r60652 = r60649 / r60651;
        return r60652;
}
↓
double f(double x) {
        double r60653 = x;
        double r60654 = exp(r60653);
        double r60655 = 0.5;
        double r60656 = 0.16666666666666666;
        double r60657 = r60656 * r60653;
        double r60658 = r60655 + r60657;
        double r60659 = r60653 * r60653;
        double r60660 = r60658 * r60659;
        double r60661 = r60653 + r60660;
        double r60662 = r60654 / r60661;
        return r60662;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.4
Target41.1
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.4

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

  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. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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