Average Error: 41.0 → 0.6
Time: 3.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.991275043797484545:\\ \;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.991275043797484545:\\
\;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\

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

\end{array}
double f(double x) {
        double r142168 = x;
        double r142169 = exp(r142168);
        double r142170 = 1.0;
        double r142171 = r142169 - r142170;
        double r142172 = r142169 / r142171;
        return r142172;
}
↓
double f(double x) {
        double r142173 = x;
        double r142174 = exp(r142173);
        double r142175 = 0.9912750437974845;
        bool r142176 = r142174 <= r142175;
        double r142177 = 1.0;
        double r142178 = -r142177;
        double r142179 = r142173 + r142173;
        double r142180 = exp(r142179);
        double r142181 = fma(r142178, r142177, r142180);
        double r142182 = r142174 + r142177;
        double r142183 = r142181 / r142182;
        double r142184 = r142174 / r142183;
        double r142185 = 0.08333333333333333;
        double r142186 = 1.0;
        double r142187 = r142186 / r142173;
        double r142188 = fma(r142185, r142173, r142187);
        double r142189 = 0.5;
        double r142190 = r142188 + r142189;
        double r142191 = r142176 ? r142184 : r142190;
        return r142191;
}

Error

Bits error versus x

Target

Original41.0
Target40.6
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9912750437974845

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Simplified0.0

      \[\leadsto \frac{e^{x}}{\frac{\color{blue}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{e^{x} + 1}}\]

    if 0.9912750437974845 < (exp x)

    1. Initial program 61.8

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

    2. Taylor expanded around 0 1.0

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]

    3. Simplified1.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.991275043797484545:\\ \;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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