Average Error: 41.4 → 0.7
Time: 9.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.00315224065076235996:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.00315224065076235996:\\
\;\;\;\;\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}\\

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

\end{array}
double f(double x) {
        double r113343 = x;
        double r113344 = exp(r113343);
        double r113345 = 1.0;
        double r113346 = r113344 - r113345;
        double r113347 = r113344 / r113346;
        return r113347;
}
↓
double f(double x) {
        double r113348 = x;
        double r113349 = exp(r113348);
        double r113350 = 0.00315224065076236;
        bool r113351 = r113349 <= r113350;
        double r113352 = sqrt(r113349);
        double r113353 = 1.0;
        double r113354 = r113349 - r113353;
        double r113355 = r113354 / r113352;
        double r113356 = r113352 / r113355;
        double r113357 = 0.08333333333333333;
        double r113358 = 1.0;
        double r113359 = r113358 / r113348;
        double r113360 = fma(r113357, r113348, r113359);
        double r113361 = 0.5;
        double r113362 = r113360 + r113361;
        double r113363 = r113351 ? r113356 : r113362;
        return r113363;
}

Error

Bits error versus x

Target

Original41.4
Target41.0
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.00315224065076236

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied associate-/l*0.0

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

    if 0.00315224065076236 < (exp x)

    1. Initial program 61.7

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.00315224065076235996:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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