Average Error: 39.7 → 0.8
Time: 18.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.450925509850113 \cdot 10^{-05}:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{3 \cdot x} + -1}{\left(e^{x} + 1\right) \cdot e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \left(\frac{1}{12} \cdot e^{\log \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)}\right) \cdot \sqrt[3]{x}\\ \end{array}\]
\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}{\left(e^{x} + 1\right) \cdot e^{x} + 1}}\\

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

\end{array}
double f(double x) {
        double r4351448 = x;
        double r4351449 = exp(r4351448);
        double r4351450 = 1.0;
        double r4351451 = r4351449 - r4351450;
        double r4351452 = r4351449 / r4351451;
        return r4351452;
}
↓
double f(double x) {
        double r4351453 = x;
        double r4351454 = -4.450925509850113e-05;
        bool r4351455 = r4351453 <= r4351454;
        double r4351456 = exp(r4351453);
        double r4351457 = 3.0;
        double r4351458 = r4351457 * r4351453;
        double r4351459 = exp(r4351458);
        double r4351460 = -1.0;
        double r4351461 = r4351459 + r4351460;
        double r4351462 = 1.0;
        double r4351463 = r4351456 + r4351462;
        double r4351464 = r4351463 * r4351456;
        double r4351465 = r4351464 + r4351462;
        double r4351466 = r4351461 / r4351465;
        double r4351467 = r4351456 / r4351466;
        double r4351468 = 0.5;
        double r4351469 = r4351462 / r4351453;
        double r4351470 = r4351468 + r4351469;
        double r4351471 = 0.08333333333333333;
        double r4351472 = cbrt(r4351453);
        double r4351473 = r4351472 * r4351472;
        double r4351474 = /* ERROR: no posit support in C */;
        double r4351475 = /* ERROR: no posit support in C */;
        double r4351476 = log(r4351475);
        double r4351477 = exp(r4351476);
        double r4351478 = r4351471 * r4351477;
        double r4351479 = r4351478 * r4351472;
        double r4351480 = r4351470 + r4351479;
        double r4351481 = r4351455 ? r4351467 : r4351480;
        return r4351481;
}

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

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

    5. Simplified0.1

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

    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

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

Reproduce

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

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

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