Average Error: 40.8 → 0.6
Time: 3.8s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le \frac{8910181223962965}{9007199254740992}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le \frac{8910181223962965}{9007199254740992}:\\
\;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\

\end{array}
double f(double x) {
        double r112189 = x;
        double r112190 = exp(r112189);
        double r112191 = 1.0;
        double r112192 = r112190 - r112191;
        double r112193 = r112190 / r112192;
        return r112193;
}
↓
double f(double x) {
        double r112194 = x;
        double r112195 = exp(r112194);
        double r112196 = 8910181223962965.0;
        double r112197 = 9007199254740992.0;
        double r112198 = r112196 / r112197;
        bool r112199 = r112195 <= r112198;
        double r112200 = 1.0;
        double r112201 = 1.0;
        double r112202 = r112201 / r112195;
        double r112203 = r112200 - r112202;
        double r112204 = r112200 / r112203;
        double r112205 = 0.5;
        double r112206 = 0.08333333333333333;
        double r112207 = r112206 * r112194;
        double r112208 = r112200 / r112194;
        double r112209 = r112207 + r112208;
        double r112210 = r112205 + r112209;
        double r112211 = r112199 ? r112204 : r112210;
        return r112211;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.8
Target40.5
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9892288348426442

    1. Initial program 0.0

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

    3. Applied clear-num0.0

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

    4. Simplified0.0

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

    if 0.9892288348426442 < (exp x)

    1. Initial program 61.8

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

    2. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le \frac{8910181223962965}{9007199254740992}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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