Average Error: 41.2 → 1.0
Time: 12.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{\frac{e^{x}}{\left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot x + 1}}{x}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{\frac{e^{x}}{\left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot x + 1}}{x}
double f(double x) {
        double r79516 = x;
        double r79517 = exp(r79516);
        double r79518 = 1.0;
        double r79519 = r79517 - r79518;
        double r79520 = r79517 / r79519;
        return r79520;
}
↓
double f(double x) {
        double r79521 = x;
        double r79522 = exp(r79521);
        double r79523 = 0.16666666666666666;
        double r79524 = r79521 * r79523;
        double r79525 = 0.5;
        double r79526 = r79524 + r79525;
        double r79527 = r79526 * r79521;
        double r79528 = 1.0;
        double r79529 = r79527 + r79528;
        double r79530 = r79522 / r79529;
        double r79531 = r79530 / r79521;
        return r79531;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.2
Target40.8
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.2

    \[\frac{e^{x}}{e^{x} - 1}\]

  2. Taylor expanded around 0 11.4

    \[\leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]

  3. Simplified1.0

    \[\leadsto \frac{e^{x}}{\color{blue}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}}\]
  4. Using strategy rm

  5. Applied clear-num1.0

    \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{e^{x}}}}\]

  6. Final simplification1.0

    \[\leadsto \frac{\frac{e^{x}}{\left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot x + 1}}{x}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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