Average Error: 29.2 → 0.0
Time: 8.3m
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.006778640856078821:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.007603939260661412:\\ \;\;\;\;\left(\left(\frac{-1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + x\right) + {x}^{5} \cdot \frac{2}{15}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - 1}{\left(1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.006778640856078821:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\mathbf{elif}\;x \le 0.007603939260661412:\\
\;\;\;\;\left(\left(\frac{-1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + x\right) + {x}^{5} \cdot \frac{2}{15}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - 1}{\left(1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r58900672 = 2.0;
        double r58900673 = 1.0;
        double r58900674 = -2.0;
        double r58900675 = x;
        double r58900676 = r58900674 * r58900675;
        double r58900677 = exp(r58900676);
        double r58900678 = r58900673 + r58900677;
        double r58900679 = r58900672 / r58900678;
        double r58900680 = r58900679 - r58900673;
        return r58900680;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r58900681 = x;
        double r58900682 = -0.006778640856078821;
        bool r58900683 = r58900681 <= r58900682;
        double r58900684 = 2.0;
        double r58900685 = 1.0;
        double r58900686 = -2.0;
        double r58900687 = r58900686 * r58900681;
        double r58900688 = exp(r58900687);
        double r58900689 = r58900685 + r58900688;
        double r58900690 = r58900684 / r58900689;
        double r58900691 = r58900690 - r58900685;
        double r58900692 = 0.007603939260661412;
        bool r58900693 = r58900681 <= r58900692;
        double r58900694 = -0.3333333333333333;
        double r58900695 = r58900681 * r58900681;
        double r58900696 = r58900694 * r58900695;
        double r58900697 = r58900696 * r58900681;
        double r58900698 = r58900697 + r58900681;
        double r58900699 = 5.0;
        double r58900700 = pow(r58900681, r58900699);
        double r58900701 = 0.13333333333333333;
        double r58900702 = r58900700 * r58900701;
        double r58900703 = r58900698 + r58900702;
        double r58900704 = r58900690 * r58900690;
        double r58900705 = 3.0;
        double r58900706 = pow(r58900704, r58900705);
        double r58900707 = r58900706 - r58900685;
        double r58900708 = r58900685 + r58900704;
        double r58900709 = r58900704 * r58900704;
        double r58900710 = r58900708 + r58900709;
        double r58900711 = r58900707 / r58900710;
        double r58900712 = r58900690 + r58900685;
        double r58900713 = r58900711 / r58900712;
        double r58900714 = r58900693 ? r58900703 : r58900713;
        double r58900715 = r58900683 ? r58900691 : r58900714;
        return r58900715;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.006778640856078821

    1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\frac{2}{e^{-2 \cdot x} + 1}} - 1\]

    if -0.006778640856078821 < x < 0.007603939260661412

    1. Initial program 58.9

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{3}\right) \cdot x + x\right) + {x}^{5} \cdot \frac{2}{15}}\]

    if 0.007603939260661412 < x

    1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\frac{2}{e^{-2 \cdot x} + 1}} - 1\]
    3. Using strategy rm

    4. Applied flip--0.0

      \[\leadsto \color{blue}{\frac{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1 \cdot 1}{\frac{2}{e^{-2 \cdot x} + 1} + 1}}\]
    5. Using strategy rm

    6. Applied flip3--0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006778640856078821:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.007603939260661412:\\ \;\;\;\;\left(\left(\frac{-1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + x\right) + {x}^{5} \cdot \frac{2}{15}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - 1}{\left(1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))