Average Error: 29.2 → 0.1
Time: 25.4s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.0646225484162213 \lor \neg \left(-2 \cdot x \le 7.02017225425806434 \cdot 10^{-6}\right):\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}} + \left(\left(-1\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -1.0646225484162213 \lor \neg \left(-2 \cdot x \le 7.02017225425806434 \cdot 10^{-6}\right):\\
\;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}} + \left(\left(-1\right) + 1\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r52360 = 2.0;
        double r52361 = 1.0;
        double r52362 = -2.0;
        double r52363 = x;
        double r52364 = r52362 * r52363;
        double r52365 = exp(r52364);
        double r52366 = r52361 + r52365;
        double r52367 = r52360 / r52366;
        double r52368 = r52367 - r52361;
        return r52368;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r52369 = -2.0;
        double r52370 = x;
        double r52371 = r52369 * r52370;
        double r52372 = -1.0646225484162213;
        bool r52373 = r52371 <= r52372;
        double r52374 = 7.020172254258064e-06;
        bool r52375 = r52371 <= r52374;
        double r52376 = !r52375;
        bool r52377 = r52373 || r52376;
        double r52378 = 2.0;
        double r52379 = 1.0;
        double r52380 = exp(r52371);
        double r52381 = r52379 + r52380;
        double r52382 = r52378 / r52381;
        double r52383 = r52382 - r52379;
        double r52384 = 3.0;
        double r52385 = pow(r52383, r52384);
        double r52386 = cbrt(r52385);
        double r52387 = -r52379;
        double r52388 = r52387 + r52379;
        double r52389 = r52386 + r52388;
        double r52390 = r52379 * r52370;
        double r52391 = 5.551115123125783e-17;
        double r52392 = 4.0;
        double r52393 = pow(r52370, r52392);
        double r52394 = 0.33333333333333337;
        double r52395 = pow(r52370, r52384);
        double r52396 = r52394 * r52395;
        double r52397 = fma(r52391, r52393, r52396);
        double r52398 = r52390 - r52397;
        double r52399 = r52377 ? r52389 : r52398;
        return r52399;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (* -2.0 x) < -1.0646225484162213 or 7.020172254258064e-06 < (* -2.0 x)

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied *-un-lft-identity0.1

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

    6. Applied times-frac0.1

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

    7. Applied prod-diff0.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

    11. Applied add-cbrt-cube0.1

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

    12. Simplified0.1

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

    if -1.0646225484162213 < (* -2.0 x) < 7.020172254258064e-06

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

      \[\leadsto \color{blue}{1 \cdot x - \mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.0646225484162213 \lor \neg \left(-2 \cdot x \le 7.02017225425806434 \cdot 10^{-6}\right):\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}} + \left(\left(-1\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))