Average Error: 41.6 → 0.2
Time: 3.2s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.997692962517644166275943007349269464612:\\ \;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)\\ \mathbf{elif}\;e^{x} \le 1.00000000133302457960837728023761883378:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \sqrt[3]{{\left(\frac{1}{e^{x}}\right)}^{3}}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.997692962517644166275943007349269464612:\\
\;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)\\

\mathbf{elif}\;e^{x} \le 1.00000000133302457960837728023761883378:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 - \sqrt[3]{{\left(\frac{1}{e^{x}}\right)}^{3}}}\\

\end{array}
double f(double x) {
        double r91373 = x;
        double r91374 = exp(r91373);
        double r91375 = 1.0;
        double r91376 = r91374 - r91375;
        double r91377 = r91374 / r91376;
        return r91377;
}
↓
double f(double x) {
        double r91378 = x;
        double r91379 = exp(r91378);
        double r91380 = 0.9976929625176442;
        bool r91381 = r91379 <= r91380;
        double r91382 = 1.0;
        double r91383 = -r91382;
        double r91384 = r91378 + r91378;
        double r91385 = exp(r91384);
        double r91386 = fma(r91383, r91382, r91385);
        double r91387 = r91379 / r91386;
        double r91388 = r91379 + r91382;
        double r91389 = r91387 * r91388;
        double r91390 = 1.0000000013330246;
        bool r91391 = r91379 <= r91390;
        double r91392 = 0.08333333333333333;
        double r91393 = 1.0;
        double r91394 = r91393 / r91378;
        double r91395 = fma(r91392, r91378, r91394);
        double r91396 = 0.5;
        double r91397 = r91395 + r91396;
        double r91398 = r91382 / r91379;
        double r91399 = 3.0;
        double r91400 = pow(r91398, r91399);
        double r91401 = cbrt(r91400);
        double r91402 = r91393 - r91401;
        double r91403 = r91393 / r91402;
        double r91404 = r91391 ? r91397 : r91403;
        double r91405 = r91381 ? r91389 : r91404;
        return r91405;
}

Error

Bits error versus x

Target

Original41.6
Target41.1
Herbie0.2
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 3 regimes

  2. if (exp x) < 0.9976929625176442

    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. Applied associate-/r/0.0

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

    5. Simplified0.0

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

    if 0.9976929625176442 < (exp x) < 1.0000000013330246

    1. Initial program 62.8

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

    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 1.0000000013330246 < (exp x)

    1. Initial program 28.3

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

    3. Applied clear-num28.3

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

    4. Simplified6.8

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

    6. Applied add-cbrt-cube7.1

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

    7. Applied add-cbrt-cube7.1

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

    8. Applied cbrt-undiv7.0

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

    9. Simplified6.9

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.997692962517644166275943007349269464612:\\ \;\;\;\;\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)\\ \mathbf{elif}\;e^{x} \le 1.00000000133302457960837728023761883378:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \sqrt[3]{{\left(\frac{1}{e^{x}}\right)}^{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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