Average Error: 41.5 → 0.7
Time: 11.2s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{e^{x}}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.0:\\
\;\;\;\;\frac{e^{x}}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)\\

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

\end{array}
double f(double x) {
        double r79197 = x;
        double r79198 = exp(r79197);
        double r79199 = 1.0;
        double r79200 = r79198 - r79199;
        double r79201 = r79198 / r79200;
        return r79201;
}
↓
double f(double x) {
        double r79202 = x;
        double r79203 = exp(r79202);
        double r79204 = 0.0;
        bool r79205 = r79203 <= r79204;
        double r79206 = r79202 + r79202;
        double r79207 = exp(r79206);
        double r79208 = 1.0;
        double r79209 = r79208 * r79208;
        double r79210 = r79207 - r79209;
        double r79211 = r79203 / r79210;
        double r79212 = r79203 + r79208;
        double r79213 = r79211 * r79212;
        double r79214 = 0.08333333333333333;
        double r79215 = 1.0;
        double r79216 = r79215 / r79202;
        double r79217 = fma(r79214, r79202, r79216);
        double r79218 = 0.5;
        double r79219 = r79217 + r79218;
        double r79220 = r79205 ? r79213 : r79219;
        return r79220;
}

Error

Bits error versus x

Target

Original41.5
Target41.2
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.0

    1. Initial program 0

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

    3. Applied flip--0

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

    4. Applied associate-/r/0

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

    5. Simplified0

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

    if 0.0 < (exp x)

    1. Initial program 61.6

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

    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 simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{e^{x}}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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