Average Error: 41.0 → 0.7
Time: 10.7s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.957041714000110066:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.957041714000110066:\\
\;\;\;\;\frac{e^{x}}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}\\

\end{array}
double f(double x) {
        double r57184 = x;
        double r57185 = exp(r57184);
        double r57186 = 1.0;
        double r57187 = r57185 - r57186;
        double r57188 = r57185 / r57187;
        return r57188;
}
↓
double f(double x) {
        double r57189 = x;
        double r57190 = exp(r57189);
        double r57191 = 0.9570417140001101;
        bool r57192 = r57190 <= r57191;
        double r57193 = r57189 + r57189;
        double r57194 = exp(r57193);
        double r57195 = 1.0;
        double r57196 = r57195 * r57195;
        double r57197 = r57194 - r57196;
        double r57198 = r57190 + r57195;
        double r57199 = r57197 / r57198;
        double r57200 = r57190 / r57199;
        double r57201 = 0.5;
        double r57202 = 2.0;
        double r57203 = pow(r57189, r57202);
        double r57204 = 0.16666666666666666;
        double r57205 = 3.0;
        double r57206 = pow(r57189, r57205);
        double r57207 = fma(r57204, r57206, r57189);
        double r57208 = fma(r57201, r57203, r57207);
        double r57209 = r57190 / r57208;
        double r57210 = r57192 ? r57200 : r57209;
        return r57210;
}

Error

Bits error versus x

Target

Original41.0
Target40.6
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9570417140001101

    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

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

    if 0.9570417140001101 < (exp x)

    1. Initial program 61.8

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

      \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.957041714000110066:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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