Average Error: 41.2 → 0.7
Time: 2.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.900874639949274303:\\ \;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.900874639949274303:\\
\;\;\;\;\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 r98121 = x;
        double r98122 = exp(r98121);
        double r98123 = 1.0;
        double r98124 = r98122 - r98123;
        double r98125 = r98122 / r98124;
        return r98125;
}
↓
double f(double x) {
        double r98126 = x;
        double r98127 = exp(r98126);
        double r98128 = 0.9008746399492743;
        bool r98129 = r98127 <= r98128;
        double r98130 = 1.0;
        double r98131 = -r98130;
        double r98132 = r98126 + r98126;
        double r98133 = exp(r98132);
        double r98134 = fma(r98131, r98130, r98133);
        double r98135 = r98127 + r98130;
        double r98136 = r98134 / r98135;
        double r98137 = r98127 / r98136;
        double r98138 = 0.08333333333333333;
        double r98139 = 1.0;
        double r98140 = r98139 / r98126;
        double r98141 = fma(r98138, r98126, r98140);
        double r98142 = 0.5;
        double r98143 = r98141 + r98142;
        double r98144 = r98129 ? r98137 : r98143;
        return r98144;
}

Error

Bits error versus x

Target

Original41.2
Target40.7
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9008746399492743

    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 0.9008746399492743 < (exp x)

    1. Initial program 61.9

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.900874639949274303:\\ \;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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