Average Error: 29.7 → 0.1
Time: 5.0s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(\frac{2}{1 + e^{-2 \cdot x}}, \frac{2}{1 + e^{-2 \cdot x}}, 1 \cdot 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\
\;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\

\mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\
\;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r57108 = 2.0;
        double r57109 = 1.0;
        double r57110 = -2.0;
        double r57111 = x;
        double r57112 = r57110 * r57111;
        double r57113 = exp(r57112);
        double r57114 = r57109 + r57113;
        double r57115 = r57108 / r57114;
        double r57116 = r57115 - r57109;
        return r57116;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r57117 = -2.0;
        double r57118 = x;
        double r57119 = r57117 * r57118;
        double r57120 = -12.329649134843827;
        bool r57121 = r57119 <= r57120;
        double r57122 = 2.0;
        double r57123 = 1.0;
        double r57124 = exp(r57119);
        double r57125 = r57123 + r57124;
        double r57126 = r57122 / r57125;
        double r57127 = r57126 * r57126;
        double r57128 = r57123 * r57123;
        double r57129 = r57127 - r57128;
        double r57130 = r57126 + r57123;
        double r57131 = r57129 / r57130;
        double r57132 = 5.306400563900346e-06;
        bool r57133 = r57119 <= r57132;
        double r57134 = 5.551115123125783e-17;
        double r57135 = 4.0;
        double r57136 = pow(r57118, r57135);
        double r57137 = 0.33333333333333337;
        double r57138 = 3.0;
        double r57139 = pow(r57118, r57138);
        double r57140 = r57137 * r57139;
        double r57141 = fma(r57134, r57136, r57140);
        double r57142 = -r57141;
        double r57143 = fma(r57123, r57118, r57142);
        double r57144 = r57127 * r57127;
        double r57145 = r57128 * r57128;
        double r57146 = r57144 - r57145;
        double r57147 = fma(r57126, r57126, r57128);
        double r57148 = r57147 * r57130;
        double r57149 = r57146 / r57148;
        double r57150 = r57133 ? r57143 : r57149;
        double r57151 = r57121 ? r57131 : r57150;
        return r57151;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (* -2.0 x) < -12.329649134843827

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    if -12.329649134843827 < (* -2.0 x) < 5.306400563900346e-06

    1. Initial program 59.0

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)}\]

    if 5.306400563900346e-06 < (* -2.0 x)

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    5. Applied flip--0.1

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

    6. Applied associate-/l/0.1

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

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(\frac{2}{1 + e^{-2 \cdot x}}, \frac{2}{1 + e^{-2 \cdot x}}, 1 \cdot 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))