Average Error: 39.4 → 1.0
Time: 14.7s
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 r2564947 = x;
        double r2564948 = exp(r2564947);
        double r2564949 = 1.0;
        double r2564950 = r2564948 - r2564949;
        double r2564951 = r2564948 / r2564950;
        return r2564951;
}
↓
double f(double x) {
        double r2564952 = x;
        double r2564953 = exp(r2564952);
        double r2564954 = 0.5;
        double r2564955 = 0.16666666666666666;
        double r2564956 = r2564955 * r2564952;
        double r2564957 = r2564954 + r2564956;
        double r2564958 = r2564952 * r2564952;
        double r2564959 = r2564957 * r2564958;
        double r2564960 = r2564952 + r2564959;
        double r2564961 = r2564953 / r2564960;
        return r2564961;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.4
Target39.0
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.4

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

  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. Simplified1.0

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

  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 2019139 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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