Average Error: 41.5 → 0.6
Time: 12.3s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0220815666007950252:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1}{1 - \frac{1}{e^{x}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.0220815666007950252:\\
\;\;\;\;\sqrt[3]{{\left(\frac{1}{1 - \frac{1}{e^{x}}}\right)}^{3}}\\

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

\end{array}
double f(double x) {
        double r91214 = x;
        double r91215 = exp(r91214);
        double r91216 = 1.0;
        double r91217 = r91215 - r91216;
        double r91218 = r91215 / r91217;
        return r91218;
}
↓
double f(double x) {
        double r91219 = x;
        double r91220 = exp(r91219);
        double r91221 = 0.022081566600795025;
        bool r91222 = r91220 <= r91221;
        double r91223 = 1.0;
        double r91224 = 1.0;
        double r91225 = r91224 / r91220;
        double r91226 = r91223 - r91225;
        double r91227 = r91223 / r91226;
        double r91228 = 3.0;
        double r91229 = pow(r91227, r91228);
        double r91230 = cbrt(r91229);
        double r91231 = 0.08333333333333333;
        double r91232 = r91223 / r91219;
        double r91233 = fma(r91231, r91219, r91232);
        double r91234 = 0.5;
        double r91235 = r91233 + r91234;
        double r91236 = r91222 ? r91230 : r91235;
        return r91236;
}

Error

Bits error versus x

Target

Original41.5
Target41.1
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.022081566600795025

    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

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{1 - \frac{1}{e^{x}}} \cdot \frac{1}{1 - \frac{1}{e^{x}}}\right) \cdot \frac{1}{1 - \frac{1}{e^{x}}}}}\]

    7. Simplified0.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{1 - \frac{1}{e^{x}}}\right)}^{3}}}\]

    if 0.022081566600795025 < (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. Simplified0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0220815666007950252:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1}{1 - \frac{1}{e^{x}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))