Average Error: 40.9 → 0.8
Time: 3.2s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 2.32657494077244701 \cdot 10^{-52}:\\ \;\;\;\;\frac{e^{x}}{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 2.32657494077244701 \cdot 10^{-52}:\\
\;\;\;\;\frac{e^{x}}{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}\\

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

\end{array}
double f(double x) {
        double r114234 = x;
        double r114235 = exp(r114234);
        double r114236 = 1.0;
        double r114237 = r114235 - r114236;
        double r114238 = r114235 / r114237;
        return r114238;
}
↓
double f(double x) {
        double r114239 = x;
        double r114240 = exp(r114239);
        double r114241 = 2.326574940772447e-52;
        bool r114242 = r114240 <= r114241;
        double r114243 = sqrt(r114240);
        double r114244 = 1.0;
        double r114245 = sqrt(r114244);
        double r114246 = r114243 + r114245;
        double r114247 = r114243 - r114245;
        double r114248 = r114246 * r114247;
        double r114249 = r114240 / r114248;
        double r114250 = 0.5;
        double r114251 = 0.08333333333333333;
        double r114252 = r114251 * r114239;
        double r114253 = 1.0;
        double r114254 = r114253 / r114239;
        double r114255 = r114252 + r114254;
        double r114256 = r114250 + r114255;
        double r114257 = r114242 ? r114249 : r114256;
        return r114257;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.9
Target40.4
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 2.326574940772447e-52

    1. Initial program 0

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

    3. Applied add-sqr-sqrt0

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

    4. Applied add-sqr-sqrt0

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

    5. Applied difference-of-squares0

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

    if 2.326574940772447e-52 < (exp x)

    1. Initial program 61.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 2.32657494077244701 \cdot 10^{-52}:\\ \;\;\;\;\frac{e^{x}}{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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