Average Error: 29.1 → 0.0
Time: 20.1s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.006778948484773374:\\ \;\;\;\;\frac{\frac{\frac{8}{e^{-2 \cdot x} + 1}}{\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)} - 1}{\frac{2}{e^{-2 \cdot x} + 1} + \mathsf{fma}\left(\frac{2}{e^{-2 \cdot x} + 1}, \frac{2}{e^{-2 \cdot x} + 1}, 1\right)}\\ \mathbf{elif}\;x \le 0.006963305059651398:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{2}{15}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{8}{e^{-2 \cdot x} + 1}}{\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)} - 1}{\frac{2}{e^{-2 \cdot x} + 1} + \mathsf{fma}\left(\frac{2}{e^{-2 \cdot x} + 1}, \frac{2}{e^{-2 \cdot x} + 1}, 1\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.006778948484773374:\\
\;\;\;\;\frac{\frac{\frac{8}{e^{-2 \cdot x} + 1}}{\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)} - 1}{\frac{2}{e^{-2 \cdot x} + 1} + \mathsf{fma}\left(\frac{2}{e^{-2 \cdot x} + 1}, \frac{2}{e^{-2 \cdot x} + 1}, 1\right)}\\

\mathbf{elif}\;x \le 0.006963305059651398:\\
\;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{2}{15}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{8}{e^{-2 \cdot x} + 1}}{\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)} - 1}{\frac{2}{e^{-2 \cdot x} + 1} + \mathsf{fma}\left(\frac{2}{e^{-2 \cdot x} + 1}, \frac{2}{e^{-2 \cdot x} + 1}, 1\right)}\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r3358553 = 2.0;
        double r3358554 = 1.0;
        double r3358555 = -2.0;
        double r3358556 = x;
        double r3358557 = r3358555 * r3358556;
        double r3358558 = exp(r3358557);
        double r3358559 = r3358554 + r3358558;
        double r3358560 = r3358553 / r3358559;
        double r3358561 = r3358560 - r3358554;
        return r3358561;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r3358562 = x;
        double r3358563 = -0.006778948484773374;
        bool r3358564 = r3358562 <= r3358563;
        double r3358565 = 8.0;
        double r3358566 = -2.0;
        double r3358567 = r3358566 * r3358562;
        double r3358568 = exp(r3358567);
        double r3358569 = 1.0;
        double r3358570 = r3358568 + r3358569;
        double r3358571 = r3358565 / r3358570;
        double r3358572 = r3358570 * r3358570;
        double r3358573 = r3358571 / r3358572;
        double r3358574 = r3358573 - r3358569;
        double r3358575 = 2.0;
        double r3358576 = r3358575 / r3358570;
        double r3358577 = fma(r3358576, r3358576, r3358569);
        double r3358578 = r3358576 + r3358577;
        double r3358579 = r3358574 / r3358578;
        double r3358580 = 0.006963305059651398;
        bool r3358581 = r3358562 <= r3358580;
        double r3358582 = 5.0;
        double r3358583 = pow(r3358562, r3358582);
        double r3358584 = 0.13333333333333333;
        double r3358585 = r3358562 * r3358562;
        double r3358586 = r3358585 * r3358562;
        double r3358587 = -0.3333333333333333;
        double r3358588 = fma(r3358586, r3358587, r3358562);
        double r3358589 = fma(r3358583, r3358584, r3358588);
        double r3358590 = r3358581 ? r3358589 : r3358579;
        double r3358591 = r3358564 ? r3358579 : r3358590;
        return r3358591;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.006778948484773374 or 0.006963305059651398 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied flip3--0.0

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

    6. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{8}{1 + e^{x \cdot -2}}}{\left(1 + e^{x \cdot -2}\right) \cdot \left(1 + e^{x \cdot -2}\right)} - 1}}{\frac{2}{1 + e^{x \cdot -2}} \cdot \frac{2}{1 + e^{x \cdot -2}} + \left(1 \cdot 1 + \frac{2}{1 + e^{x \cdot -2}} \cdot 1\right)}\]

    7. Simplified0.0

      \[\leadsto \frac{\frac{\frac{8}{1 + e^{x \cdot -2}}}{\left(1 + e^{x \cdot -2}\right) \cdot \left(1 + e^{x \cdot -2}\right)} - 1}{\color{blue}{\frac{2}{1 + e^{x \cdot -2}} + \mathsf{fma}\left(\frac{2}{1 + e^{x \cdot -2}}, \frac{2}{1 + e^{x \cdot -2}}, 1\right)}}\]

    if -0.006778948484773374 < x < 0.006963305059651398

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006778948484773374:\\ \;\;\;\;\frac{\frac{\frac{8}{e^{-2 \cdot x} + 1}}{\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)} - 1}{\frac{2}{e^{-2 \cdot x} + 1} + \mathsf{fma}\left(\frac{2}{e^{-2 \cdot x} + 1}, \frac{2}{e^{-2 \cdot x} + 1}, 1\right)}\\ \mathbf{elif}\;x \le 0.006963305059651398:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{2}{15}, \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{8}{e^{-2 \cdot x} + 1}}{\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)} - 1}{\frac{2}{e^{-2 \cdot x} + 1} + \mathsf{fma}\left(\frac{2}{e^{-2 \cdot x} + 1}, \frac{2}{e^{-2 \cdot x} + 1}, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))