Average Error: 40.4 → 0.3
Time: 32.3s
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 r2180905 = x;
        double r2180906 = exp(r2180905);
        double r2180907 = 1.0;
        double r2180908 = r2180906 - r2180907;
        double r2180909 = r2180906 / r2180908;
        return r2180909;
}
↓
double f(double x) {
        double r2180910 = x;
        double r2180911 = exp(r2180910);
        double r2180912 = expm1(r2180910);
        double r2180913 = r2180911 / r2180912;
        return r2180913;
}

Error

Bits error versus x

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Results

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Target

Original40.4
Target40.0
Herbie0.3
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.4

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

  3. Applied expm1-log1p-u40.4

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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