Average Error: 40.6 → 0.0
Time: 3.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9972044245763829195539074135012924671173:\\ \;\;\;\;\frac{\frac{1}{1 + \sqrt{\frac{1}{e^{x}}}}}{\sqrt{1} - \sqrt{\frac{1}{e^{x}}}}\\ \mathbf{elif}\;e^{x} \le 1.00021080983059862390405214682687073946:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\log \left(e^{1 - \frac{1}{e^{x}}}\right)}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9972044245763829195539074135012924671173:\\
\;\;\;\;\frac{\frac{1}{1 + \sqrt{\frac{1}{e^{x}}}}}{\sqrt{1} - \sqrt{\frac{1}{e^{x}}}}\\

\mathbf{elif}\;e^{x} \le 1.00021080983059862390405214682687073946:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\log \left(e^{1 - \frac{1}{e^{x}}}\right)}\\

\end{array}
double f(double x) {
        double r75500 = x;
        double r75501 = exp(r75500);
        double r75502 = 1.0;
        double r75503 = r75501 - r75502;
        double r75504 = r75501 / r75503;
        return r75504;
}
↓
double f(double x) {
        double r75505 = x;
        double r75506 = exp(r75505);
        double r75507 = 0.9972044245763829;
        bool r75508 = r75506 <= r75507;
        double r75509 = 1.0;
        double r75510 = 1.0;
        double r75511 = r75510 / r75506;
        double r75512 = sqrt(r75511);
        double r75513 = r75509 + r75512;
        double r75514 = r75509 / r75513;
        double r75515 = sqrt(r75509);
        double r75516 = r75515 - r75512;
        double r75517 = r75514 / r75516;
        double r75518 = 1.0002108098305986;
        bool r75519 = r75506 <= r75518;
        double r75520 = 0.08333333333333333;
        double r75521 = r75509 / r75505;
        double r75522 = fma(r75520, r75505, r75521);
        double r75523 = 0.5;
        double r75524 = r75522 + r75523;
        double r75525 = r75509 - r75511;
        double r75526 = exp(r75525);
        double r75527 = log(r75526);
        double r75528 = r75509 / r75527;
        double r75529 = r75519 ? r75524 : r75528;
        double r75530 = r75508 ? r75517 : r75529;
        return r75530;
}

Error

Bits error versus x

Target

Original40.6
Target40.2
Herbie0.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 3 regimes

  2. if (exp x) < 0.9972044245763829

    1. Initial program 0.0

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

    3. Applied clear-num0.0

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

    4. Simplified0.0

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied add-sqr-sqrt0.0

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

    8. Applied difference-of-squares0.0

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

    9. Applied associate-/r*0.0

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

    10. Simplified0.0

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

    if 0.9972044245763829 < (exp x) < 1.0002108098305986

    1. Initial program 62.4

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

    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 1.0002108098305986 < (exp x)

    1. Initial program 32.9

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

    3. Applied clear-num32.9

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

    4. Simplified1.5

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

    6. Applied add-log-exp1.6

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

    7. Applied add-log-exp1.6

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

    8. Applied diff-log1.7

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

    9. Simplified1.7

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9972044245763829195539074135012924671173:\\ \;\;\;\;\frac{\frac{1}{1 + \sqrt{\frac{1}{e^{x}}}}}{\sqrt{1} - \sqrt{\frac{1}{e^{x}}}}\\ \mathbf{elif}\;e^{x} \le 1.00021080983059862390405214682687073946:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\log \left(e^{1 - \frac{1}{e^{x}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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