Average Error: 41.6 → 1.1
Time: 2.6s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{\sqrt[3]{e^{x}}}}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{\sqrt[3]{e^{x}}}}
double f(double x) {
        double r88156 = x;
        double r88157 = exp(r88156);
        double r88158 = 1.0;
        double r88159 = r88157 - r88158;
        double r88160 = r88157 / r88159;
        return r88160;
}
↓
double f(double x) {
        double r88161 = x;
        double r88162 = exp(r88161);
        double r88163 = cbrt(r88162);
        double r88164 = r88163 * r88163;
        double r88165 = 2.0;
        double r88166 = pow(r88161, r88165);
        double r88167 = 0.16666666666666666;
        double r88168 = r88161 * r88167;
        double r88169 = 0.5;
        double r88170 = r88168 + r88169;
        double r88171 = r88166 * r88170;
        double r88172 = r88171 + r88161;
        double r88173 = r88172 / r88163;
        double r88174 = r88164 / r88173;
        return r88174;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.6
Target41.2
Herbie1.1
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.6

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

  2. Taylor expanded around 0 11.8

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

  3. Simplified1.1

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

  5. Applied add-cube-cbrt1.1

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

  6. Applied associate-/l*1.1

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

  7. Final simplification1.1

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

Reproduce

herbie shell --seed 2020021 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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