Average Error: 40.3 → 0.0
Time: 4.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.3453084686247354 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x}}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\\ \mathbf{elif}\;x \le 0.0017789298620281822:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}\\ \end{array}\]
\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{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}\\

\end{array}
double f(double x) {
        double r369 = x;
        double r370 = exp(r369);
        double r371 = 1.0;
        double r372 = r370 - r371;
        double r373 = r370 / r372;
        return r373;
}
↓
double f(double x) {
        double r374 = x;
        double r375 = -0.0007345308468624735;
        bool r376 = r374 <= r375;
        double r377 = exp(r374);
        double r378 = 1.0;
        double r379 = -r378;
        double r380 = r374 + r374;
        double r381 = exp(r380);
        double r382 = fma(r379, r378, r381);
        double r383 = r377 + r378;
        double r384 = r382 / r383;
        double r385 = r377 / r384;
        double r386 = 0.0017789298620281822;
        bool r387 = r374 <= r386;
        double r388 = 0.08333333333333333;
        double r389 = 1.0;
        double r390 = r389 / r374;
        double r391 = fma(r388, r374, r390);
        double r392 = 0.5;
        double r393 = r391 + r392;
        double r394 = log(r378);
        double r395 = r394 - r374;
        double r396 = exp(r395);
        double r397 = r389 - r396;
        double r398 = cbrt(r397);
        double r399 = r389 / r398;
        double r400 = r399 / r398;
        double r401 = r400 / r398;
        double r402 = r387 ? r393 : r401;
        double r403 = r376 ? r385 : r402;
        return r403;
}

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 associate-/r*0.7

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

    11. Simplified0.7

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

  4. Final simplification0.0

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

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)))