Average Error: 39.6 → 0.4
Time: 11.6s
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 r1468056 = x;
        double r1468057 = exp(r1468056);
        double r1468058 = 1.0;
        double r1468059 = r1468057 - r1468058;
        double r1468060 = r1468057 / r1468059;
        return r1468060;
}
↓
double f(double x) {
        double r1468061 = x;
        double r1468062 = exp(r1468061);
        double r1468063 = expm1(r1468061);
        double r1468064 = r1468062 / r1468063;
        return r1468064;
}

Error

Bits error versus x

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Results

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Target

Original39.6
Target39.2
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.6

    \[\frac{e^{x}}{e^{x} - 1}\]
  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 2019152 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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