Average Error: 40.0 → 0.7
Time: 28.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.7019935717068964:\\ \;\;\;\;\frac{e^{x}}{\frac{\frac{\frac{-1 + \left(e^{3 \cdot x} \cdot e^{3 \cdot x}\right) \cdot e^{3 \cdot x}}{\left(e^{3 \cdot x} \cdot e^{3 \cdot x} + 1\right) + e^{3 \cdot x}}}{\sqrt{\left(1 + e^{x}\right) \cdot e^{x} + 1}}}{\sqrt{\left(1 + e^{x}\right) \cdot e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.7019935717068964:\\
\;\;\;\;\frac{e^{x}}{\frac{\frac{\frac{-1 + \left(e^{3 \cdot x} \cdot e^{3 \cdot x}\right) \cdot e^{3 \cdot x}}{\left(e^{3 \cdot x} \cdot e^{3 \cdot x} + 1\right) + e^{3 \cdot x}}}{\sqrt{\left(1 + e^{x}\right) \cdot e^{x} + 1}}}{\sqrt{\left(1 + e^{x}\right) \cdot e^{x} + 1}}}\\

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

\end{array}
double f(double x) {
        double r3046159 = x;
        double r3046160 = exp(r3046159);
        double r3046161 = 1.0;
        double r3046162 = r3046160 - r3046161;
        double r3046163 = r3046160 / r3046162;
        return r3046163;
}
↓
double f(double x) {
        double r3046164 = x;
        double r3046165 = exp(r3046164);
        double r3046166 = 0.7019935717068964;
        bool r3046167 = r3046165 <= r3046166;
        double r3046168 = -1.0;
        double r3046169 = 3.0;
        double r3046170 = r3046169 * r3046164;
        double r3046171 = exp(r3046170);
        double r3046172 = r3046171 * r3046171;
        double r3046173 = r3046172 * r3046171;
        double r3046174 = r3046168 + r3046173;
        double r3046175 = 1.0;
        double r3046176 = r3046172 + r3046175;
        double r3046177 = r3046176 + r3046171;
        double r3046178 = r3046174 / r3046177;
        double r3046179 = r3046175 + r3046165;
        double r3046180 = r3046179 * r3046165;
        double r3046181 = r3046180 + r3046175;
        double r3046182 = sqrt(r3046181);
        double r3046183 = r3046178 / r3046182;
        double r3046184 = r3046183 / r3046182;
        double r3046185 = r3046165 / r3046184;
        double r3046186 = r3046175 / r3046164;
        double r3046187 = 0.5;
        double r3046188 = r3046186 + r3046187;
        double r3046189 = 0.08333333333333333;
        double r3046190 = r3046189 * r3046164;
        double r3046191 = r3046188 + r3046190;
        double r3046192 = r3046167 ? r3046185 : r3046191;
        return r3046192;
}

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. Simplified0.0

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

    5. Simplified0.0

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

    7. Applied add-sqr-sqrt0.0

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

    8. Applied associate-/r*0.0

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

    10. Applied flip3-+0.0

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

    11. Simplified0.0

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

    12. Simplified0.0

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

    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

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

Reproduce

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

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

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