Average Error: 39.7 → 0.5
Time: 14.8s
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 r3272990 = x;
        double r3272991 = exp(r3272990);
        double r3272992 = 1.0;
        double r3272993 = r3272991 - r3272992;
        double r3272994 = r3272991 / r3272993;
        return r3272994;
}
↓
double f(double x) {
        double r3272995 = x;
        double r3272996 = exp(r3272995);
        double r3272997 = expm1(r3272995);
        double r3272998 = r3272996 / r3272997;
        return r3272998;
}

Error

Bits error versus x

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Results

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Target

Original39.7
Target39.3
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.7

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

  3. Applied expm1-log1p-u39.7

    \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} - 1\right)\right)}}\]

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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