Average Error: 29.4 → 0.1
Time: 35.6s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.3535994530314286454775185575272189453244:\\ \;\;\;\;\mathsf{fma}\left(1, -1, 1\right) + \left(\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\right)\\ \mathbf{elif}\;-2 \cdot x \le 1.360762104871435039234140937369765822496 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot x - \left(x \cdot 0.3333333333333333703407674875052180141211 + \left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, -1, 1\right) + \left(\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.3535994530314286454775185575272189453244:\\
\;\;\;\;\mathsf{fma}\left(1, -1, 1\right) + \left(\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\right)\\

\mathbf{elif}\;-2 \cdot x \le 1.360762104871435039234140937369765822496 \cdot 10^{-6}:\\
\;\;\;\;1 \cdot x - \left(x \cdot 0.3333333333333333703407674875052180141211 + \left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right) \cdot \left(x \cdot x\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r3078725 = 2.0;
        double r3078726 = 1.0;
        double r3078727 = -2.0;
        double r3078728 = x;
        double r3078729 = r3078727 * r3078728;
        double r3078730 = exp(r3078729);
        double r3078731 = r3078726 + r3078730;
        double r3078732 = r3078725 / r3078731;
        double r3078733 = r3078732 - r3078726;
        return r3078733;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r3078734 = -2.0;
        double r3078735 = x;
        double r3078736 = r3078734 * r3078735;
        double r3078737 = -0.35359945303142865;
        bool r3078738 = r3078736 <= r3078737;
        double r3078739 = 1.0;
        double r3078740 = -1.0;
        double r3078741 = fma(r3078739, r3078740, r3078739);
        double r3078742 = 2.0;
        double r3078743 = exp(r3078736);
        double r3078744 = r3078743 + r3078739;
        double r3078745 = sqrt(r3078744);
        double r3078746 = r3078742 / r3078745;
        double r3078747 = r3078746 / r3078745;
        double r3078748 = r3078747 - r3078739;
        double r3078749 = r3078741 + r3078748;
        double r3078750 = 1.360762104871435e-06;
        bool r3078751 = r3078736 <= r3078750;
        double r3078752 = r3078739 * r3078735;
        double r3078753 = 0.33333333333333337;
        double r3078754 = r3078735 * r3078753;
        double r3078755 = r3078735 * r3078735;
        double r3078756 = 5.551115123125783e-17;
        double r3078757 = r3078755 * r3078756;
        double r3078758 = r3078754 + r3078757;
        double r3078759 = r3078758 * r3078755;
        double r3078760 = r3078752 - r3078759;
        double r3078761 = r3078751 ? r3078760 : r3078749;
        double r3078762 = r3078738 ? r3078749 : r3078761;
        return r3078762;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (* -2.0 x) < -0.35359945303142865 or 1.360762104871435e-06 < (* -2.0 x)

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied add-sqr-sqrt0.9

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

    6. Applied times-frac0.9

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

    7. Applied prod-diff0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}}, -\sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{1}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)}\]

    8. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\right)} + \mathsf{fma}\left(-\sqrt[3]{1}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)\]

    9. Simplified0.1

      \[\leadsto \left(\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\right) + \color{blue}{\mathsf{fma}\left(1, -1, 1\right)}\]

    if -0.35359945303142865 < (* -2.0 x) < 1.360762104871435e-06

    1. Initial program 59.3

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{1 \cdot x - \left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4}\right)}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{x \cdot 1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333703407674875052180141211 \cdot x + \left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.3535994530314286454775185575272189453244:\\ \;\;\;\;\mathsf{fma}\left(1, -1, 1\right) + \left(\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\right)\\ \mathbf{elif}\;-2 \cdot x \le 1.360762104871435039234140937369765822496 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot x - \left(x \cdot 0.3333333333333333703407674875052180141211 + \left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, -1, 1\right) + \left(\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))