Average Error: 41.4 → 0.7
Time: 11.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.00315224065076235996:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.00315224065076235996:\\
\;\;\;\;\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}\\

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

\end{array}
double f(double x) {
        double r98137 = x;
        double r98138 = exp(r98137);
        double r98139 = 1.0;
        double r98140 = r98138 - r98139;
        double r98141 = r98138 / r98140;
        return r98141;
}
↓
double f(double x) {
        double r98142 = x;
        double r98143 = exp(r98142);
        double r98144 = 0.00315224065076236;
        bool r98145 = r98143 <= r98144;
        double r98146 = sqrt(r98143);
        double r98147 = 1.0;
        double r98148 = r98143 - r98147;
        double r98149 = r98148 / r98146;
        double r98150 = r98146 / r98149;
        double r98151 = 0.08333333333333333;
        double r98152 = 1.0;
        double r98153 = r98152 / r98142;
        double r98154 = fma(r98151, r98142, r98153);
        double r98155 = 0.5;
        double r98156 = r98154 + r98155;
        double r98157 = r98145 ? r98150 : r98156;
        return r98157;
}

Error

Bits error versus x

Target

Original41.4
Target41.0
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.00315224065076236

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied associate-/l*0.0

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

    if 0.00315224065076236 < (exp x)

    1. Initial program 61.7

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.00315224065076235996:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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