Average Error: 40.1 → 0.4
Time: 14.6s
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 r1229977 = x;
        double r1229978 = exp(r1229977);
        double r1229979 = 1.0;
        double r1229980 = r1229978 - r1229979;
        double r1229981 = r1229978 / r1229980;
        return r1229981;
}
↓
double f(double x) {
        double r1229982 = x;
        double r1229983 = exp(r1229982);
        double r1229984 = expm1(r1229982);
        double r1229985 = r1229983 / r1229984;
        return r1229985;
}

Error

Bits error versus x

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Results

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Target

Original40.1
Target39.7
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.1

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

  3. Applied expm1-log1p-u40.1

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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