Average Error: 41.5 → 0.9
Time: 13.0s
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 r6524874 = x;
        double r6524875 = exp(r6524874);
        double r6524876 = 1.0;
        double r6524877 = r6524875 - r6524876;
        double r6524878 = r6524875 / r6524877;
        return r6524878;
}
↓
double f(double x) {
        double r6524879 = x;
        double r6524880 = exp(r6524879);
        double r6524881 = 0.5;
        double r6524882 = 0.16666666666666666;
        double r6524883 = r6524882 * r6524879;
        double r6524884 = r6524881 + r6524883;
        double r6524885 = r6524879 * r6524879;
        double r6524886 = r6524884 * r6524885;
        double r6524887 = r6524879 + r6524886;
        double r6524888 = r6524880 / r6524887;
        return r6524888;
}

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

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

  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)))