Average Error: 41.5 → 0.0
Time: 10.5s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00178468728852752530136815956041118624853:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x + x}}{1 + e^{x}} - \frac{1 \cdot 1}{1 + e^{x}}}\\ \mathbf{elif}\;x \le 0.001716590725359118732584740030233660945669:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \frac{1}{\frac{e^{x + x}}{1}}}{1 + \frac{1}{e^{x}}}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00178468728852752530136815956041118624853:\\
\;\;\;\;\frac{e^{x}}{\frac{e^{x + x}}{1 + e^{x}} - \frac{1 \cdot 1}{1 + e^{x}}}\\

\mathbf{elif}\;x \le 0.001716590725359118732584740030233660945669:\\
\;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{1 - \frac{1}{\frac{e^{x + x}}{1}}}{1 + \frac{1}{e^{x}}}}\\

\end{array}
double f(double x) {
        double r71503 = x;
        double r71504 = exp(r71503);
        double r71505 = 1.0;
        double r71506 = r71504 - r71505;
        double r71507 = r71504 / r71506;
        return r71507;
}
↓
double f(double x) {
        double r71508 = x;
        double r71509 = -0.0017846872885275253;
        bool r71510 = r71508 <= r71509;
        double r71511 = exp(r71508);
        double r71512 = r71508 + r71508;
        double r71513 = exp(r71512);
        double r71514 = 1.0;
        double r71515 = r71514 + r71511;
        double r71516 = r71513 / r71515;
        double r71517 = r71514 * r71514;
        double r71518 = r71517 / r71515;
        double r71519 = r71516 - r71518;
        double r71520 = r71511 / r71519;
        double r71521 = 0.0017165907253591187;
        bool r71522 = r71508 <= r71521;
        double r71523 = 0.5;
        double r71524 = 0.08333333333333333;
        double r71525 = r71524 * r71508;
        double r71526 = 1.0;
        double r71527 = r71526 / r71508;
        double r71528 = r71525 + r71527;
        double r71529 = r71523 + r71528;
        double r71530 = r71513 / r71514;
        double r71531 = r71514 / r71530;
        double r71532 = r71526 - r71531;
        double r71533 = r71514 / r71511;
        double r71534 = r71526 + r71533;
        double r71535 = r71532 / r71534;
        double r71536 = r71526 / r71535;
        double r71537 = r71522 ? r71529 : r71536;
        double r71538 = r71510 ? r71520 : r71537;
        return r71538;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.5
Target41.0
Herbie0.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.0017846872885275253

    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

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

    5. Simplified0.0

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

    7. Applied div-sub0.0

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

    if -0.0017846872885275253 < x < 0.0017165907253591187

    1. Initial program 62.4

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

    2. Taylor expanded around 0 0.0

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

    if 0.0017165907253591187 < x

    1. Initial program 38.8

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

    3. Applied clear-num38.8

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

    4. Simplified0.5

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

    6. Applied flip--0.4

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

    7. Simplified0.3

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00178468728852752530136815956041118624853:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x + x}}{1 + e^{x}} - \frac{1 \cdot 1}{1 + e^{x}}}\\ \mathbf{elif}\;x \le 0.001716590725359118732584740030233660945669:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \frac{1}{\frac{e^{x + x}}{1}}}{1 + \frac{1}{e^{x}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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