Average Error: 40.0 → 0.7
Time: 15.8s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.7019935717068964:\\ \;\;\;\;\frac{e^{x}}{\left(\sqrt[3]{e^{x + \left(x + x\right)} - 1} \cdot \sqrt[3]{e^{x + \left(x + x\right)} - 1}\right) \cdot \sqrt[3]{\left(\sqrt{e^{x + \left(x + x\right)}} - 1\right) \cdot \left(\sqrt{e^{x + \left(x + x\right)}} + 1\right)}} \cdot \left(\left(1 + e^{x}\right) + e^{x} \cdot e^{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.7019935717068964:\\
\;\;\;\;\frac{e^{x}}{\left(\sqrt[3]{e^{x + \left(x + x\right)} - 1} \cdot \sqrt[3]{e^{x + \left(x + x\right)} - 1}\right) \cdot \sqrt[3]{\left(\sqrt{e^{x + \left(x + x\right)}} - 1\right) \cdot \left(\sqrt{e^{x + \left(x + x\right)}} + 1\right)}} \cdot \left(\left(1 + e^{x}\right) + e^{x} \cdot e^{x}\right)\\

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

\end{array}
double f(double x) {
        double r1879412 = x;
        double r1879413 = exp(r1879412);
        double r1879414 = 1.0;
        double r1879415 = r1879413 - r1879414;
        double r1879416 = r1879413 / r1879415;
        return r1879416;
}
↓
double f(double x) {
        double r1879417 = x;
        double r1879418 = exp(r1879417);
        double r1879419 = 0.7019935717068964;
        bool r1879420 = r1879418 <= r1879419;
        double r1879421 = r1879417 + r1879417;
        double r1879422 = r1879417 + r1879421;
        double r1879423 = exp(r1879422);
        double r1879424 = 1.0;
        double r1879425 = r1879423 - r1879424;
        double r1879426 = cbrt(r1879425);
        double r1879427 = r1879426 * r1879426;
        double r1879428 = sqrt(r1879423);
        double r1879429 = r1879428 - r1879424;
        double r1879430 = r1879428 + r1879424;
        double r1879431 = r1879429 * r1879430;
        double r1879432 = cbrt(r1879431);
        double r1879433 = r1879427 * r1879432;
        double r1879434 = r1879418 / r1879433;
        double r1879435 = r1879424 + r1879418;
        double r1879436 = r1879418 * r1879418;
        double r1879437 = r1879435 + r1879436;
        double r1879438 = r1879434 * r1879437;
        double r1879439 = 0.08333333333333333;
        double r1879440 = r1879417 * r1879439;
        double r1879441 = 0.5;
        double r1879442 = r1879424 / r1879417;
        double r1879443 = r1879441 + r1879442;
        double r1879444 = r1879440 + r1879443;
        double r1879445 = r1879420 ? r1879438 : r1879444;
        return r1879445;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target39.6
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.7019935717068964

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

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

    7. Applied add-cube-cbrt0.0

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

    9. Applied *-un-lft-identity0.0

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

    10. Applied add-sqr-sqrt0.0

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

    11. Applied difference-of-squares0.0

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

    if 0.7019935717068964 < (exp x)

    1. Initial program 59.9

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

    2. Taylor expanded around 0 1.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.7019935717068964:\\ \;\;\;\;\frac{e^{x}}{\left(\sqrt[3]{e^{x + \left(x + x\right)} - 1} \cdot \sqrt[3]{e^{x + \left(x + x\right)} - 1}\right) \cdot \sqrt[3]{\left(\sqrt{e^{x + \left(x + x\right)}} - 1\right) \cdot \left(\sqrt{e^{x + \left(x + x\right)}} + 1\right)}} \cdot \left(\left(1 + e^{x}\right) + e^{x} \cdot e^{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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