Average Error: 39.7 → 0.8
Time: 17.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.450925509850113 \cdot 10^{-05}:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{3 \cdot x} - 1}{1 + e^{x} \cdot \left(e^{x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \sqrt[3]{x} \cdot \left(e^{\log \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)} \cdot \frac{1}{12}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.450925509850113 \cdot 10^{-05}:\\
\;\;\;\;\frac{e^{x}}{\frac{e^{3 \cdot x} - 1}{1 + e^{x} \cdot \left(e^{x} + 1\right)}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \sqrt[3]{x} \cdot \left(e^{\log \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)} \cdot \frac{1}{12}\right)\\

\end{array}
double f(double x) {
        double r4375773 = x;
        double r4375774 = exp(r4375773);
        double r4375775 = 1.0;
        double r4375776 = r4375774 - r4375775;
        double r4375777 = r4375774 / r4375776;
        return r4375777;
}
↓
double f(double x) {
        double r4375778 = x;
        double r4375779 = -4.450925509850113e-05;
        bool r4375780 = r4375778 <= r4375779;
        double r4375781 = exp(r4375778);
        double r4375782 = 3.0;
        double r4375783 = r4375782 * r4375778;
        double r4375784 = exp(r4375783);
        double r4375785 = 1.0;
        double r4375786 = r4375784 - r4375785;
        double r4375787 = r4375781 + r4375785;
        double r4375788 = r4375781 * r4375787;
        double r4375789 = r4375785 + r4375788;
        double r4375790 = r4375786 / r4375789;
        double r4375791 = r4375781 / r4375790;
        double r4375792 = 0.5;
        double r4375793 = r4375785 / r4375778;
        double r4375794 = r4375792 + r4375793;
        double r4375795 = cbrt(r4375778);
        double r4375796 = r4375795 * r4375795;
        double r4375797 = /* ERROR: no posit support in C */;
        double r4375798 = /* ERROR: no posit support in C */;
        double r4375799 = log(r4375798);
        double r4375800 = exp(r4375799);
        double r4375801 = 0.08333333333333333;
        double r4375802 = r4375800 * r4375801;
        double r4375803 = r4375795 * r4375802;
        double r4375804 = r4375794 + r4375803;
        double r4375805 = r4375780 ? r4375791 : r4375804;
        return r4375805;
}

Error

Bits error versus x

Target

Original39.7
Target39.3
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4.450925509850113e-05

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

    if -4.450925509850113e-05 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 1.1

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt1.1

      \[\leadsto \frac{1}{12} \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} + \left(\frac{1}{x} + \frac{1}{2}\right)\]

    5. Applied associate-*r*1.1

      \[\leadsto \color{blue}{\left(\frac{1}{12} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}} + \left(\frac{1}{x} + \frac{1}{2}\right)\]
    6. Using strategy rm

    7. Applied insert-posit161.1

      \[\leadsto \left(\frac{1}{12} \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}\right) \cdot \sqrt[3]{x} + \left(\frac{1}{x} + \frac{1}{2}\right)\]
    8. Using strategy rm

    9. Applied add-exp-log1.1

      \[\leadsto \left(\frac{1}{12} \cdot \color{blue}{e^{\log \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)}}\right) \cdot \sqrt[3]{x} + \left(\frac{1}{x} + \frac{1}{2}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.450925509850113 \cdot 10^{-05}:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{3 \cdot x} - 1}{1 + e^{x} \cdot \left(e^{x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \sqrt[3]{x} \cdot \left(e^{\log \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)} \cdot \frac{1}{12}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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