Average Error: 41.5 → 0.7
Time: 3.5s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.8526603959516880770763691543834283947945:\\ \;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.8526603959516880770763691543834283947945:\\
\;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(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 r101520 = x;
        double r101521 = exp(r101520);
        double r101522 = 1.0;
        double r101523 = r101521 - r101522;
        double r101524 = r101521 / r101523;
        return r101524;
}
↓
double f(double x) {
        double r101525 = x;
        double r101526 = exp(r101525);
        double r101527 = 0.8526603959516881;
        bool r101528 = r101526 <= r101527;
        double r101529 = 1.0;
        double r101530 = -r101529;
        double r101531 = r101525 + r101525;
        double r101532 = exp(r101531);
        double r101533 = fma(r101530, r101529, r101532);
        double r101534 = r101526 / r101533;
        double r101535 = r101526 + r101529;
        double r101536 = r101534 * r101535;
        double r101537 = 0.08333333333333333;
        double r101538 = 1.0;
        double r101539 = r101538 / r101525;
        double r101540 = fma(r101537, r101525, r101539);
        double r101541 = 0.5;
        double r101542 = r101540 + r101541;
        double r101543 = r101528 ? r101536 : r101542;
        return r101543;
}

Error

Bits error versus x

Target

Original41.5
Target41.0
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.8526603959516881

    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. Applied associate-/r/0.0

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

    5. Simplified0.0

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

    if 0.8526603959516881 < (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

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.8526603959516880770763691543834283947945:\\ \;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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