Average Error: 41.5 → 0.9
Time: 14.6s
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 r3148804 = x;
        double r3148805 = exp(r3148804);
        double r3148806 = 1.0;
        double r3148807 = r3148805 - r3148806;
        double r3148808 = r3148805 / r3148807;
        return r3148808;
}
↓
double f(double x) {
        double r3148809 = x;
        double r3148810 = exp(r3148809);
        double r3148811 = 0.5;
        double r3148812 = 0.16666666666666666;
        double r3148813 = r3148812 * r3148809;
        double r3148814 = r3148811 + r3148813;
        double r3148815 = r3148809 * r3148809;
        double r3148816 = r3148814 * r3148815;
        double r3148817 = r3148809 + r3148816;
        double r3148818 = r3148810 / r3148817;
        return r3148818;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.5
Target41.2
Herbie0.9
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.5

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

  2. Taylor expanded around 0 11.4

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

  3. Simplified0.9

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

  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 2019171 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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