Average Error: 40.0 → 0.4
Time: 19.3s
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 r2978413 = x;
        double r2978414 = exp(r2978413);
        double r2978415 = 1.0;
        double r2978416 = r2978414 - r2978415;
        double r2978417 = r2978414 / r2978416;
        return r2978417;
}
↓
double f(double x) {
        double r2978418 = x;
        double r2978419 = exp(r2978418);
        double r2978420 = expm1(r2978418);
        double r2978421 = r2978419 / r2978420;
        return r2978421;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-log1p-u40.0

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

  4. Simplified0.4

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

  5. 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)))