Average Error: 29.4 → 0.2
Time: 6.9s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -5.111290763866047193175745633197948336601:\\ \;\;\;\;\sqrt[3]{{\left(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \le 3.978856656911251737182447316826250371946 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -5.111290763866047193175745633197948336601:\\
\;\;\;\;\sqrt[3]{{\left(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)\right)}^{3}}\\

\mathbf{elif}\;-2 \cdot x \le 3.978856656911251737182447316826250371946 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r68434 = 2.0;
        double r68435 = 1.0;
        double r68436 = -2.0;
        double r68437 = x;
        double r68438 = r68436 * r68437;
        double r68439 = exp(r68438);
        double r68440 = r68435 + r68439;
        double r68441 = r68434 / r68440;
        double r68442 = r68441 - r68435;
        return r68442;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r68443 = -2.0;
        double r68444 = x;
        double r68445 = r68443 * r68444;
        double r68446 = -5.111290763866047;
        bool r68447 = r68445 <= r68446;
        double r68448 = 2.0;
        double r68449 = 1.0;
        double r68450 = exp(r68445);
        double r68451 = r68449 + r68450;
        double r68452 = r68448 / r68451;
        double r68453 = sqrt(r68452);
        double r68454 = sqrt(r68449);
        double r68455 = r68453 + r68454;
        double r68456 = r68453 - r68454;
        double r68457 = r68455 * r68456;
        double r68458 = 3.0;
        double r68459 = pow(r68457, r68458);
        double r68460 = cbrt(r68459);
        double r68461 = 3.978856656911252e-07;
        bool r68462 = r68445 <= r68461;
        double r68463 = 5.551115123125783e-17;
        double r68464 = 4.0;
        double r68465 = pow(r68444, r68464);
        double r68466 = 0.33333333333333337;
        double r68467 = pow(r68444, r68458);
        double r68468 = r68466 * r68467;
        double r68469 = fma(r68463, r68465, r68468);
        double r68470 = -r68469;
        double r68471 = fma(r68449, r68444, r68470);
        double r68472 = r68452 - r68449;
        double r68473 = pow(r68472, r68458);
        double r68474 = cbrt(r68473);
        double r68475 = r68462 ? r68471 : r68474;
        double r68476 = r68447 ? r68460 : r68475;
        return r68476;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (* -2.0 x) < -5.111290763866047

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.0

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

    4. Simplified0.0

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied add-sqr-sqrt2.0

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

    8. Applied difference-of-squares0.0

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

    if -5.111290763866047 < (* -2.0 x) < 3.978856656911252e-07

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 3.978856656911252e-07 < (* -2.0 x)

    1. Initial program 0.2

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

    3. Applied add-cbrt-cube0.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -5.111290763866047193175745633197948336601:\\ \;\;\;\;\sqrt[3]{{\left(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \le 3.978856656911251737182447316826250371946 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))