Average Error: 41.5 → 1.0
Time: 3.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}
double f(double x) {
        double r115301 = x;
        double r115302 = exp(r115301);
        double r115303 = 1.0;
        double r115304 = r115302 - r115303;
        double r115305 = r115302 / r115304;
        return r115305;
}
↓
double f(double x) {
        double r115306 = x;
        double r115307 = exp(r115306);
        double r115308 = 2.0;
        double r115309 = pow(r115306, r115308);
        double r115310 = 0.16666666666666666;
        double r115311 = r115306 * r115310;
        double r115312 = 0.5;
        double r115313 = r115311 + r115312;
        double r115314 = r115309 * r115313;
        double r115315 = r115314 + r115306;
        double r115316 = r115307 / r115315;
        return r115316;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.5
Target41.1
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.5

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

  2. Taylor expanded around 0 11.6

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

  3. Simplified1.0

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

  4. Final simplification1.0

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

Reproduce

herbie shell --seed 2019356 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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