Average Error: 41.3 → 0.7
Time: 17.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9920261382877882949671288770332466810942:\\ \;\;\;\;\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9920261382877882949671288770332466810942:\\
\;\;\;\;\frac{e^{x}}{\log \left(e^{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 r35035 = x;
        double r35036 = exp(r35035);
        double r35037 = 1.0;
        double r35038 = r35036 - r35037;
        double r35039 = r35036 / r35038;
        return r35039;
}
↓
double f(double x) {
        double r35040 = x;
        double r35041 = exp(r35040);
        double r35042 = 0.9920261382877883;
        bool r35043 = r35041 <= r35042;
        double r35044 = 1.0;
        double r35045 = r35041 - r35044;
        double r35046 = exp(r35045);
        double r35047 = log(r35046);
        double r35048 = r35041 / r35047;
        double r35049 = 0.08333333333333333;
        double r35050 = 1.0;
        double r35051 = r35050 / r35040;
        double r35052 = fma(r35049, r35040, r35051);
        double r35053 = 0.5;
        double r35054 = r35052 + r35053;
        double r35055 = r35043 ? r35048 : r35054;
        return r35055;
}

Error

Bits error versus x

Target

Original41.3
Target40.8
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9920261382877883

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    if 0.9920261382877883 < (exp x)

    1. Initial program 62.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9920261382877882949671288770332466810942:\\ \;\;\;\;\frac{e^{x}}{\log \left(e^{e^{x} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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