Average Error: 40.8 → 0.5
Time: 13.7s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x} + 1}{\mathsf{expm1}\left(x + x\right)} \cdot e^{x}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x} + 1}{\mathsf{expm1}\left(x + x\right)} \cdot e^{x}
double f(double x) {
        double r4162989 = x;
        double r4162990 = exp(r4162989);
        double r4162991 = 1.0;
        double r4162992 = r4162990 - r4162991;
        double r4162993 = r4162990 / r4162992;
        return r4162993;
}
↓
double f(double x) {
        double r4162994 = x;
        double r4162995 = exp(r4162994);
        double r4162996 = 1.0;
        double r4162997 = r4162995 + r4162996;
        double r4162998 = r4162994 + r4162994;
        double r4162999 = expm1(r4162998);
        double r4163000 = r4162997 / r4162999;
        double r4163001 = r4163000 * r4162995;
        return r4163001;
}

Error

Bits error versus x

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Results

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Target

Original40.8
Target40.4
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.8

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

  3. Applied flip--40.8

    \[\leadsto \frac{e^{x}}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}\]

  4. Applied associate-/r/40.8

    \[\leadsto \color{blue}{\frac{e^{x}}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\]

  5. Simplified0.5

    \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x + x\right)}} \cdot \left(e^{x} + 1\right)\]
  6. Using strategy rm

  7. Applied div-inv0.5

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

  8. Applied associate-*l*0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

    \[\leadsto \frac{e^{x} + 1}{\mathsf{expm1}\left(x + x\right)} \cdot e^{x}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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