Average Error: 41.7 → 0.8
Time: 11.5s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 + \frac{\sqrt{1}}{\sqrt{e^{x}}}} \cdot \frac{1}{1 - \frac{\sqrt{1}}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.0:\\
\;\;\;\;\frac{1}{1 + \frac{\sqrt{1}}{\sqrt{e^{x}}}} \cdot \frac{1}{1 - \frac{\sqrt{1}}{\sqrt{e^{x}}}}\\

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

\end{array}
double f(double x) {
        double r79585 = x;
        double r79586 = exp(r79585);
        double r79587 = 1.0;
        double r79588 = r79586 - r79587;
        double r79589 = r79586 / r79588;
        return r79589;
}
↓
double f(double x) {
        double r79590 = x;
        double r79591 = exp(r79590);
        double r79592 = 0.0;
        bool r79593 = r79591 <= r79592;
        double r79594 = 1.0;
        double r79595 = 1.0;
        double r79596 = sqrt(r79595);
        double r79597 = sqrt(r79591);
        double r79598 = r79596 / r79597;
        double r79599 = r79594 + r79598;
        double r79600 = r79594 / r79599;
        double r79601 = r79594 - r79598;
        double r79602 = r79594 / r79601;
        double r79603 = r79600 * r79602;
        double r79604 = 0.5;
        double r79605 = 0.08333333333333333;
        double r79606 = r79605 * r79590;
        double r79607 = r79594 / r79590;
        double r79608 = r79606 + r79607;
        double r79609 = r79604 + r79608;
        double r79610 = r79593 ? r79603 : r79609;
        return r79610;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.7
Target41.3
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.0

    1. Initial program 0

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

    3. Applied clear-num0

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

    4. Simplified0

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

    6. Applied add-sqr-sqrt0

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

    7. Applied add-sqr-sqrt0

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

    8. Applied times-frac0

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

    9. Applied add-sqr-sqrt0

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

    10. Applied difference-of-squares0

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

    11. Applied add-cube-cbrt0

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

    12. Applied times-frac0

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

    13. Simplified0

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

    14. Simplified0

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

    if 0.0 < (exp x)

    1. Initial program 61.6

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 + \frac{\sqrt{1}}{\sqrt{e^{x}}}} \cdot \frac{1}{1 - \frac{\sqrt{1}}{\sqrt{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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