Average Error: 40.3 → 0.6
Time: 11.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.002322584501578668:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \mathsf{fma}\left(\frac{1}{12}, x, \left(\frac{1}{x}\right)\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.002322584501578668:\\
\;\;\;\;\frac{1}{1 - e^{-x}}\\

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

\end{array}
double f(double x) {
        double r1399639 = x;
        double r1399640 = exp(r1399639);
        double r1399641 = 1.0;
        double r1399642 = r1399640 - r1399641;
        double r1399643 = r1399640 / r1399642;
        return r1399643;
}
↓
double f(double x) {
        double r1399644 = x;
        double r1399645 = -0.002322584501578668;
        bool r1399646 = r1399644 <= r1399645;
        double r1399647 = 1.0;
        double r1399648 = -r1399644;
        double r1399649 = exp(r1399648);
        double r1399650 = r1399647 - r1399649;
        double r1399651 = r1399647 / r1399650;
        double r1399652 = 0.5;
        double r1399653 = 0.08333333333333333;
        double r1399654 = r1399647 / r1399644;
        double r1399655 = fma(r1399653, r1399644, r1399654);
        double r1399656 = r1399652 + r1399655;
        double r1399657 = r1399646 ? r1399651 : r1399656;
        return r1399657;
}

Error

Bits error versus x

Target

Original40.3
Target39.9
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.002322584501578668

    1. Initial program 0.0

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

    3. Applied expm1-log1p-u0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(e^{x} - 1\right)\right)\right)\right)}}\]

    4. Simplified0.0

      \[\leadsto \frac{e^{x}}{\mathsf{expm1}\left(\color{blue}{x}\right)}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity0.0

      \[\leadsto \frac{\color{blue}{1 \cdot e^{x}}}{\mathsf{expm1}\left(x\right)}\]

    7. Applied associate-/l*0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}}\]

    8. Simplified0.0

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

    if -0.002322584501578668 < x

    1. Initial program 60.3

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

    2. Taylor expanded around 0 0.9

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

    3. Simplified0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.002322584501578668:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \mathsf{fma}\left(\frac{1}{12}, x, \left(\frac{1}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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