Average Error: 41.0 → 0.6
Time: 19.5s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.4621431440290150738370300587121164426208:\\ \;\;\;\;\frac{e^{x}}{{\left(e^{x}\right)}^{3} - {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.4621431440290150738370300587121164426208:\\
\;\;\;\;\frac{e^{x}}{{\left(e^{x}\right)}^{3} - {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\

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

\end{array}
double f(double x) {
        double r74837 = x;
        double r74838 = exp(r74837);
        double r74839 = 1.0;
        double r74840 = r74838 - r74839;
        double r74841 = r74838 / r74840;
        return r74841;
}
↓
double f(double x) {
        double r74842 = x;
        double r74843 = exp(r74842);
        double r74844 = 0.4621431440290151;
        bool r74845 = r74843 <= r74844;
        double r74846 = 3.0;
        double r74847 = pow(r74843, r74846);
        double r74848 = 1.0;
        double r74849 = pow(r74848, r74846);
        double r74850 = r74847 - r74849;
        double r74851 = r74843 / r74850;
        double r74852 = r74843 * r74843;
        double r74853 = r74848 * r74848;
        double r74854 = r74843 * r74848;
        double r74855 = r74853 + r74854;
        double r74856 = r74852 + r74855;
        double r74857 = r74851 * r74856;
        double r74858 = 0.5;
        double r74859 = 0.08333333333333333;
        double r74860 = 1.0;
        double r74861 = r74860 / r74842;
        double r74862 = fma(r74859, r74842, r74861);
        double r74863 = r74858 + r74862;
        double r74864 = r74845 ? r74857 : r74863;
        return r74864;
}

Error

Bits error versus x

Target

Original41.0
Target40.6
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.4621431440290151

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Applied associate-/r/0.0

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

    if 0.4621431440290151 < (exp x)

    1. Initial program 61.7

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]

    3. Simplified1.2

      \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)}}\]

    4. Taylor expanded around 0 0.9

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

    5. Simplified0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.4621431440290150738370300587121164426208:\\ \;\;\;\;\frac{e^{x}}{{\left(e^{x}\right)}^{3} - {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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