Average Error: 40.0 → 0.4
Time: 8.9s
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 r1814611 = x;
        double r1814612 = exp(r1814611);
        double r1814613 = 1.0;
        double r1814614 = r1814612 - r1814613;
        double r1814615 = r1814612 / r1814614;
        return r1814615;
}
↓
double f(double x) {
        double r1814616 = x;
        double r1814617 = exp(r1814616);
        double r1814618 = expm1(r1814616);
        double r1814619 = r1814617 / r1814618;
        return r1814619;
}

Error

Bits error versus x

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Results

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Target

Original40.0
Target39.6
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.0

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

  3. Applied expm1-def0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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