Average Error: 40.1 → 0.4
Time: 12.6s
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 r3808613 = x;
        double r3808614 = exp(r3808613);
        double r3808615 = 1.0;
        double r3808616 = r3808614 - r3808615;
        double r3808617 = r3808614 / r3808616;
        return r3808617;
}
↓
double f(double x) {
        double r3808618 = x;
        double r3808619 = exp(r3808618);
        double r3808620 = expm1(r3808618);
        double r3808621 = r3808619 / r3808620;
        return r3808621;
}

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-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 2019120 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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