Average Error: 39.2 → 0.6
Time: 25.3s
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 r2719580 = x;
        double r2719581 = exp(r2719580);
        double r2719582 = 1.0;
        double r2719583 = r2719581 - r2719582;
        double r2719584 = r2719581 / r2719583;
        return r2719584;
}
↓
double f(double x) {
        double r2719585 = x;
        double r2719586 = exp(r2719585);
        double r2719587 = expm1(r2719585);
        double r2719588 = r2719586 / r2719587;
        return r2719588;
}

Error

Bits error versus x

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Results

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Target

Original39.2
Target38.7
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.2

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

  3. Applied expm1-log1p-u39.2

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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