Average Error: 41.4 → 0.6
Time: 3.2s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.94364060397973859029008281140704639256:\\ \;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.94364060397973859029008281140704639256:\\
\;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\

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

\end{array}
double f(double x) {
        double r107378 = x;
        double r107379 = exp(r107378);
        double r107380 = 1.0;
        double r107381 = r107379 - r107380;
        double r107382 = r107379 / r107381;
        return r107382;
}
↓
double f(double x) {
        double r107383 = x;
        double r107384 = exp(r107383);
        double r107385 = 0.9436406039797386;
        bool r107386 = r107384 <= r107385;
        double r107387 = 1.0;
        double r107388 = -r107387;
        double r107389 = r107383 + r107383;
        double r107390 = exp(r107389);
        double r107391 = fma(r107388, r107387, r107390);
        double r107392 = r107384 + r107387;
        double r107393 = r107391 / r107392;
        double r107394 = r107384 / r107393;
        double r107395 = 0.08333333333333333;
        double r107396 = 1.0;
        double r107397 = r107396 / r107383;
        double r107398 = fma(r107395, r107383, r107397);
        double r107399 = 0.5;
        double r107400 = r107398 + r107399;
        double r107401 = r107386 ? r107394 : r107400;
        return r107401;
}

Error

Bits error versus x

Target

Original41.4
Target41.0
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9436406039797386

    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. Simplified0.0

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

    if 0.9436406039797386 < (exp x)

    1. Initial program 61.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.94364060397973859029008281140704639256:\\ \;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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