Average Error: 39.4 → 0.4
Time: 17.4s
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 r1883268 = x;
        double r1883269 = exp(r1883268);
        double r1883270 = 1.0;
        double r1883271 = r1883269 - r1883270;
        double r1883272 = r1883269 / r1883271;
        return r1883272;
}
↓
double f(double x) {
        double r1883273 = x;
        double r1883274 = exp(r1883273);
        double r1883275 = expm1(r1883273);
        double r1883276 = r1883274 / r1883275;
        return r1883276;
}

Error

Bits error versus x

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Results

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Target

Original39.4
Target39.0
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.4

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

  3. Applied expm1-log1p-u39.4

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

  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 2019139 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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