Average Error: 40.6 → 1.1
Time: 14.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}
double f(double x) {
        double r3279951 = x;
        double r3279952 = exp(r3279951);
        double r3279953 = 1.0;
        double r3279954 = r3279952 - r3279953;
        double r3279955 = r3279952 / r3279954;
        return r3279955;
}
↓
double f(double x) {
        double r3279956 = x;
        double r3279957 = exp(r3279956);
        double r3279958 = 0.5;
        double r3279959 = 0.16666666666666666;
        double r3279960 = r3279959 * r3279956;
        double r3279961 = r3279958 + r3279960;
        double r3279962 = r3279956 * r3279956;
        double r3279963 = r3279961 * r3279962;
        double r3279964 = r3279956 + r3279963;
        double r3279965 = r3279957 / r3279964;
        return r3279965;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.6
Target40.1
Herbie1.1
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.6

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

  2. Taylor expanded around 0 11.4

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

  3. Simplified1.1

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

  4. Final simplification1.1

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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