Average Error: 41.2 → 1.0
Time: 13.8s
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 r64142 = x;
        double r64143 = exp(r64142);
        double r64144 = 1.0;
        double r64145 = r64143 - r64144;
        double r64146 = r64143 / r64145;
        return r64146;
}
↓
double f(double x) {
        double r64147 = x;
        double r64148 = exp(r64147);
        double r64149 = 0.5;
        double r64150 = 0.16666666666666666;
        double r64151 = r64150 * r64147;
        double r64152 = r64149 + r64151;
        double r64153 = r64147 * r64147;
        double r64154 = r64152 * r64153;
        double r64155 = r64147 + r64154;
        double r64156 = r64148 / r64155;
        return r64156;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original41.2
Target40.9
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.2

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

  2. Taylor expanded around 0 11.5

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

  3. Simplified1.0

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

  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 2019195 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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