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 r69452 = 2.0;
        double r69453 = 1.0;
        double r69454 = -2.0;
        double r69455 = x;
        double r69456 = r69454 * r69455;
        double r69457 = exp(r69456);
        double r69458 = r69453 + r69457;
        double r69459 = r69452 / r69458;
        double r69460 = r69459 - r69453;
        return r69460;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r69461 = -2.0;
        double r69462 = x;
        double r69463 = r69461 * r69462;
        double r69464 = -5.111290763866047;
        bool r69465 = r69463 <= r69464;
        double r69466 = 2.0;
        double r69467 = 1.0;
        double r69468 = exp(r69463);
        double r69469 = r69467 + r69468;
        double r69470 = r69466 / r69469;
        double r69471 = sqrt(r69470);
        double r69472 = sqrt(r69467);
        double r69473 = r69471 + r69472;
        double r69474 = r69471 - r69472;
        double r69475 = r69473 * r69474;
        double r69476 = 3.0;
        double r69477 = pow(r69475, r69476);
        double r69478 = cbrt(r69477);
        double r69479 = 3.978856656911252e-07;
        bool r69480 = r69463 <= r69479;
        double r69481 = 5.551115123125783e-17;
        double r69482 = 4.0;
        double r69483 = pow(r69462, r69482);
        double r69484 = 0.33333333333333337;
        double r69485 = pow(r69462, r69476);
        double r69486 = r69484 * r69485;
        double r69487 = fma(r69481, r69483, r69486);
        double r69488 = -r69487;
        double r69489 = fma(r69467, r69462, r69488);
        double r69490 = r69470 - r69467;
        double r69491 = pow(r69490, r69476);
        double r69492 = cbrt(r69491);
        double r69493 = r69480 ? r69489 : r69492;
        double r69494 = r69465 ? r69478 : r69493;
        return r69494;
}

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))