Average Error: 28.8 → 0.5
Time: 13.2s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -4690010.71037716139 \lor \neg \left(-2 \cdot x \le 3.2942790551493054 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\right) + \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x - \mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right) + \left(1 - 1\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -4690010.71037716139 \lor \neg \left(-2 \cdot x \le 3.2942790551493054 \cdot 10^{-5}\right):\\
\;\;\;\;\left(\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\right) + \left(1 - 1\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r36260 = 2.0;
        double r36261 = 1.0;
        double r36262 = -2.0;
        double r36263 = x;
        double r36264 = r36262 * r36263;
        double r36265 = exp(r36264);
        double r36266 = r36261 + r36265;
        double r36267 = r36260 / r36266;
        double r36268 = r36267 - r36261;
        return r36268;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r36269 = -2.0;
        double r36270 = x;
        double r36271 = r36269 * r36270;
        double r36272 = -4690010.710377161;
        bool r36273 = r36271 <= r36272;
        double r36274 = 3.2942790551493054e-05;
        bool r36275 = r36271 <= r36274;
        double r36276 = !r36275;
        bool r36277 = r36273 || r36276;
        double r36278 = 2.0;
        double r36279 = 1.0;
        double r36280 = exp(r36271);
        double r36281 = r36279 + r36280;
        double r36282 = sqrt(r36281);
        double r36283 = r36278 / r36282;
        double r36284 = r36283 / r36282;
        double r36285 = r36284 - r36279;
        double r36286 = r36279 - r36279;
        double r36287 = r36285 + r36286;
        double r36288 = r36279 * r36270;
        double r36289 = 5.551115123125783e-17;
        double r36290 = 4.0;
        double r36291 = pow(r36270, r36290);
        double r36292 = 0.33333333333333337;
        double r36293 = 3.0;
        double r36294 = pow(r36270, r36293);
        double r36295 = r36292 * r36294;
        double r36296 = fma(r36289, r36291, r36295);
        double r36297 = r36288 - r36296;
        double r36298 = r36297 + r36286;
        double r36299 = r36277 ? r36287 : r36298;
        return r36299;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (* -2.0 x) < -4690010.710377161 or 3.2942790551493054e-05 < (* -2.0 x)

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    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 add-sqr-sqrt0.8

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

    6. Applied times-frac0.8

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

    7. Applied prod-diff0.5

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

    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

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

    if -4690010.710377161 < (* -2.0 x) < 3.2942790551493054e-05

    1. Initial program 58.5

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

    3. Applied add-cube-cbrt58.5

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

    4. Applied add-sqr-sqrt59.4

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

    5. Applied add-sqr-sqrt58.6

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

    6. Applied times-frac58.6

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

    7. Applied prod-diff58.6

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

    8. Simplified59.4

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

    9. Simplified59.4

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

    10. Taylor expanded around 0 0.9

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

    11. Simplified0.9

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -4690010.71037716139 \lor \neg \left(-2 \cdot x \le 3.2942790551493054 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\right) + \left(1 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x - \mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right) + \left(1 - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))