Average Error: 41.2 → 1.0
Time: 30.1s
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 r5331730 = x;
        double r5331731 = exp(r5331730);
        double r5331732 = 1.0;
        double r5331733 = r5331731 - r5331732;
        double r5331734 = r5331731 / r5331733;
        return r5331734;
}
↓
double f(double x) {
        double r5331735 = x;
        double r5331736 = exp(r5331735);
        double r5331737 = 0.5;
        double r5331738 = 0.16666666666666666;
        double r5331739 = r5331738 * r5331735;
        double r5331740 = r5331737 + r5331739;
        double r5331741 = r5331735 * r5331735;
        double r5331742 = r5331740 * r5331741;
        double r5331743 = r5331735 + r5331742;
        double r5331744 = r5331736 / r5331743;
        return r5331744;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Simplified1.0

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

  4. Final simplification1.0

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

Reproduce

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

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

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