Average Error: 40.6 → 0.4
Time: 3.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 1.000000952951165533733046686393208801746:\\ \;\;\;\;\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 1.000000952951165533733046686393208801746:\\
\;\;\;\;\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\

\end{array}
double f(double x) {
        double r112959 = x;
        double r112960 = exp(r112959);
        double r112961 = 1.0;
        double r112962 = r112960 - r112961;
        double r112963 = r112960 / r112962;
        return r112963;
}
↓
double f(double x) {
        double r112964 = x;
        double r112965 = exp(r112964);
        double r112966 = 1.0000009529511655;
        bool r112967 = r112965 <= r112966;
        double r112968 = 2.0;
        double r112969 = pow(r112964, r112968);
        double r112970 = 0.16666666666666666;
        double r112971 = r112964 * r112970;
        double r112972 = 0.5;
        double r112973 = r112971 + r112972;
        double r112974 = r112969 * r112973;
        double r112975 = r112974 + r112964;
        double r112976 = r112965 / r112975;
        double r112977 = 1.0;
        double r112978 = 1.0;
        double r112979 = r112978 / r112965;
        double r112980 = r112977 - r112979;
        double r112981 = r112977 / r112980;
        double r112982 = r112967 ? r112976 : r112981;
        return r112982;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.6
Target40.2
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 1.0000009529511655

    1. Initial program 40.7

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

    2. Taylor expanded around 0 11.5

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

    3. Simplified0.4

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

    if 1.0000009529511655 < (exp x)

    1. Initial program 29.6

      \[\frac{e^{x}}{e^{x} - 1}\]
    2. Using strategy rm

    3. Applied clear-num29.6

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]

    4. Simplified3.9

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 1.000000952951165533733046686393208801746:\\ \;\;\;\;\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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