Average Error: 40.3 → 0.0
Time: 3.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.3453084686247354 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\ \mathbf{elif}\;x \le 0.0017789298620281822:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}} \cdot \frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.3453084686247354 \cdot 10^{-4}:\\
\;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\

\mathbf{elif}\;x \le 0.0017789298620281822:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}} \cdot \frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}\\

\end{array}
double f(double x) {
        double r107550 = x;
        double r107551 = exp(r107550);
        double r107552 = 1.0;
        double r107553 = r107551 - r107552;
        double r107554 = r107551 / r107553;
        return r107554;
}
↓
double f(double x) {
        double r107555 = x;
        double r107556 = -0.0007345308468624735;
        bool r107557 = r107555 <= r107556;
        double r107558 = exp(r107555);
        double r107559 = 1.0;
        double r107560 = -r107559;
        double r107561 = r107555 + r107555;
        double r107562 = exp(r107561);
        double r107563 = fma(r107560, r107559, r107562);
        double r107564 = r107558 + r107559;
        double r107565 = r107563 / r107564;
        double r107566 = r107558 / r107565;
        double r107567 = 0.0017789298620281822;
        bool r107568 = r107555 <= r107567;
        double r107569 = 0.08333333333333333;
        double r107570 = 1.0;
        double r107571 = r107570 / r107555;
        double r107572 = fma(r107569, r107555, r107571);
        double r107573 = 0.5;
        double r107574 = r107572 + r107573;
        double r107575 = log(r107559);
        double r107576 = r107575 - r107555;
        double r107577 = exp(r107576);
        double r107578 = r107570 - r107577;
        double r107579 = cbrt(r107578);
        double r107580 = r107570 / r107579;
        double r107581 = r107580 / r107579;
        double r107582 = r107581 * r107580;
        double r107583 = r107568 ? r107574 : r107582;
        double r107584 = r107557 ? r107566 : r107583;
        return r107584;
}

Error

Bits error versus x

Target

Original40.3
Target39.8
Herbie0.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.0007345308468624735

    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 -0.0007345308468624735 < x < 0.0017789298620281822

    1. Initial program 62.3

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 0.0017789298620281822 < x

    1. Initial program 37.2

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

    3. Applied clear-num37.2

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

    4. Simplified0.8

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
    5. Using strategy rm

    6. Applied add-exp-log0.8

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

    7. Applied div-exp0.5

      \[\leadsto \frac{1}{1 - \color{blue}{e^{\log 1 - x}}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.7

      \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{1 - e^{\log 1 - x}} \cdot \sqrt[3]{1 - e^{\log 1 - x}}\right) \cdot \sqrt[3]{1 - e^{\log 1 - x}}}}\]

    10. Applied add-cube-cbrt0.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{1 - e^{\log 1 - x}} \cdot \sqrt[3]{1 - e^{\log 1 - x}}\right) \cdot \sqrt[3]{1 - e^{\log 1 - x}}}\]

    11. Applied times-frac0.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{1 - e^{\log 1 - x}} \cdot \sqrt[3]{1 - e^{\log 1 - x}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{1 - e^{\log 1 - x}}}}\]

    12. Simplified0.7

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{1 - e^{\log 1 - x}}}\]

    13. Simplified0.7

      \[\leadsto \frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}} \cdot \color{blue}{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.3453084686247354 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\ \mathbf{elif}\;x \le 0.0017789298620281822:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}} \cdot \frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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