Average Error: 39.9 → 0.4
Time: 12.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{\mathsf{expm1}\left(x\right)}
double f(double x) {
        double r2830714 = x;
        double r2830715 = exp(r2830714);
        double r2830716 = 1.0;
        double r2830717 = r2830715 - r2830716;
        double r2830718 = r2830715 / r2830717;
        return r2830718;
}
↓
double f(double x) {
        double r2830719 = x;
        double r2830720 = exp(r2830719);
        double r2830721 = expm1(r2830719);
        double r2830722 = r2830720 / r2830721;
        return r2830722;
}

Error

Bits error versus x

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Results

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Target

Original39.9
Target39.4
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.9

    \[\frac{e^{x}}{e^{x} - 1}\]
  2. Using strategy rm

  3. Applied expm1-def0.4

    \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}}\]

  4. Final simplification0.4

    \[\leadsto \frac{e^{x}}{\mathsf{expm1}\left(x\right)}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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