Average Error: 29.3 → 0.1
Time: 33.3s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -2.9497819558082474:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;-2 \cdot x \le 0.009141633682236621:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, -1\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, -1\right)}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -2.9497819558082474:\\
\;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\

\mathbf{elif}\;-2 \cdot x \le 0.009141633682236621:\\
\;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, -1\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, -1\right)}\right)\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r2020102 = 2.0;
        double r2020103 = 1.0;
        double r2020104 = -2.0;
        double r2020105 = x;
        double r2020106 = r2020104 * r2020105;
        double r2020107 = exp(r2020106);
        double r2020108 = r2020103 + r2020107;
        double r2020109 = r2020102 / r2020108;
        double r2020110 = r2020109 - r2020103;
        return r2020110;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r2020111 = -2.0;
        double r2020112 = x;
        double r2020113 = r2020111 * r2020112;
        double r2020114 = -2.9497819558082474;
        bool r2020115 = r2020113 <= r2020114;
        double r2020116 = 2.0;
        double r2020117 = exp(r2020113);
        double r2020118 = 1.0;
        double r2020119 = r2020117 + r2020118;
        double r2020120 = r2020116 / r2020119;
        double r2020121 = r2020120 - r2020118;
        double r2020122 = 0.009141633682236621;
        bool r2020123 = r2020113 <= r2020122;
        double r2020124 = r2020112 * r2020112;
        double r2020125 = r2020112 * r2020124;
        double r2020126 = -0.3333333333333333;
        double r2020127 = 5.0;
        double r2020128 = pow(r2020112, r2020127);
        double r2020129 = 0.13333333333333333;
        double r2020130 = fma(r2020128, r2020129, r2020112);
        double r2020131 = fma(r2020125, r2020126, r2020130);
        double r2020132 = cbrt(r2020120);
        double r2020133 = r2020132 * r2020132;
        double r2020134 = -1.0;
        double r2020135 = fma(r2020133, r2020132, r2020134);
        double r2020136 = cbrt(r2020135);
        double r2020137 = r2020136 * r2020136;
        double r2020138 = r2020136 * r2020137;
        double r2020139 = r2020123 ? r2020131 : r2020138;
        double r2020140 = r2020115 ? r2020121 : r2020139;
        return r2020140;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (* -2 x) < -2.9497819558082474

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt2.3

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

    4. Applied fma-neg2.6

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

    5. Simplified2.6

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -2.9497819558082474 < (* -2 x) < 0.009141633682236621

    1. Initial program 59.0

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

    2. Taylor expanded around -inf 59.0

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

    3. Taylor expanded around 0 0.1

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

    4. Simplified0.1

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

    if 0.009141633682236621 < (* -2 x)

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    4. Applied fma-neg0.0

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

    5. Simplified0.0

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

    7. Applied add-cube-cbrt0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -2.9497819558082474:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;-2 \cdot x \le 0.009141633682236621:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, -1\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1}}, -1\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))