Average Error: 41.0 → 0.7
Time: 9.9s
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 r69095 = x;
        double r69096 = exp(r69095);
        double r69097 = 1.0;
        double r69098 = r69096 - r69097;
        double r69099 = r69096 / r69098;
        return r69099;
}
↓
double f(double x) {
        double r69100 = x;
        double r69101 = exp(r69100);
        double r69102 = 0.9570417140001101;
        bool r69103 = r69101 <= r69102;
        double r69104 = r69100 + r69100;
        double r69105 = exp(r69104);
        double r69106 = 1.0;
        double r69107 = r69106 * r69106;
        double r69108 = r69105 - r69107;
        double r69109 = r69101 + r69106;
        double r69110 = r69108 / r69109;
        double r69111 = r69101 / r69110;
        double r69112 = 0.5;
        double r69113 = 2.0;
        double r69114 = pow(r69100, r69113);
        double r69115 = 0.16666666666666666;
        double r69116 = 3.0;
        double r69117 = pow(r69100, r69116);
        double r69118 = fma(r69115, r69117, r69100);
        double r69119 = fma(r69112, r69114, r69118);
        double r69120 = r69101 / r69119;
        double r69121 = r69103 ? r69111 : r69120;
        return r69121;
}

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)))