Logistic function from Lakshay Garg

Average Error: 29.1 → 0.2

Time: 36.2s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot -2 \le -0.1837147540782742005660566064761951565742:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{2}{e^{x \cdot -2} + 1} \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) - \left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(\frac{2}{e^{x \cdot -2} + 1} \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) - \left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1} - 1 \cdot 1\right)\right)}}{\sqrt[3]{\left(\left(1 \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right) + \frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) \cdot \left(1 \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right) + \frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right)\right) \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right)}}\\ \mathbf{elif}\;x \cdot -2 \le 1.042492423760650159831312601586774009566 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(x \cdot x\right) + 0.3333333333333333703407674875052180141211 \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{2}{e^{x \cdot -2} + 1} \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) - \left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(\frac{2}{e^{x \cdot -2} + 1} \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) - \left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1} - 1 \cdot 1\right)\right)}}{\sqrt[3]{\left(\left(1 \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right) + \frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) \cdot \left(1 \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right) + \frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right)\right) \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right)}}\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r3135548 = 2.0;
        double r3135549 = 1.0;
        double r3135550 = -2.0;
        double r3135551 = x;
        double r3135552 = r3135550 * r3135551;
        double r3135553 = exp(r3135552);
        double r3135554 = r3135549 + r3135553;
        double r3135555 = r3135548 / r3135554;
        double r3135556 = r3135555 - r3135549;
        return r3135556;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r3135557 = x;
        double r3135558 = -2.0;
        double r3135559 = r3135557 * r3135558;
        double r3135560 = -0.1837147540782742;
        bool r3135561 = r3135559 <= r3135560;
        double r3135562 = 2.0;
        double r3135563 = exp(r3135559);
        double r3135564 = 1.0;
        double r3135565 = r3135563 + r3135564;
        double r3135566 = r3135562 / r3135565;
        double r3135567 = r3135566 * r3135566;
        double r3135568 = r3135566 * r3135567;
        double r3135569 = r3135564 * r3135564;
        double r3135570 = r3135569 * r3135564;
        double r3135571 = r3135568 - r3135570;
        double r3135572 = r3135567 - r3135569;
        double r3135573 = r3135571 * r3135572;
        double r3135574 = r3135571 * r3135573;
        double r3135575 = cbrt(r3135574);
        double r3135576 = r3135564 + r3135566;
        double r3135577 = r3135564 * r3135576;
        double r3135578 = r3135577 + r3135567;
        double r3135579 = r3135578 * r3135578;
        double r3135580 = r3135579 * r3135576;
        double r3135581 = cbrt(r3135580);
        double r3135582 = r3135575 / r3135581;
        double r3135583 = 1.0424924237606502e-10;
        bool r3135584 = r3135559 <= r3135583;
        double r3135585 = r3135564 * r3135557;
        double r3135586 = 5.551115123125783e-17;
        double r3135587 = r3135557 * r3135557;
        double r3135588 = r3135586 * r3135587;
        double r3135589 = 0.33333333333333337;
        double r3135590 = r3135589 * r3135557;
        double r3135591 = r3135588 + r3135590;
        double r3135592 = r3135591 * r3135587;
        double r3135593 = r3135585 - r3135592;
        double r3135594 = r3135584 ? r3135593 : r3135582;
        double r3135595 = r3135561 ? r3135582 : r3135594;
        return r3135595;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if (* -2.0 x) < -0.1837147540782742 or 1.0424924237606502e-10 < (* -2.0 x)

   1. Initial program 0.2

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

   3. Applied add-cbrt-cube 0.2

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

   5. Applied flip-- 0.2

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

   6. Applied flip3-- 0.2

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

   7. Applied flip3-- 0.2

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

   8. Applied frac-times 0.2

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

   9. Applied frac-times 0.2

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

   10. Applied cbrt-div 0.2

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

   11. Simplified 0.2

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

   12. Simplified 0.2

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

   if -0.1837147540782742 < (* -2.0 x) < 1.0424924237606502e-10

   1. Initial program 59.4

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

   2. Taylor expanded around 0 0.1

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{x \cdot 1 - \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333703407674875052180141211 + \left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \le -0.1837147540782742005660566064761951565742:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{2}{e^{x \cdot -2} + 1} \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) - \left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(\frac{2}{e^{x \cdot -2} + 1} \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) - \left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1} - 1 \cdot 1\right)\right)}}{\sqrt[3]{\left(\left(1 \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right) + \frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) \cdot \left(1 \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right) + \frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right)\right) \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right)}}\\ \mathbf{elif}\;x \cdot -2 \le 1.042492423760650159831312601586774009566 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(x \cdot x\right) + 0.3333333333333333703407674875052180141211 \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{2}{e^{x \cdot -2} + 1} \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) - \left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(\frac{2}{e^{x \cdot -2} + 1} \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) - \left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1} - 1 \cdot 1\right)\right)}}{\sqrt[3]{\left(\left(1 \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right) + \frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right) \cdot \left(1 \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right) + \frac{2}{e^{x \cdot -2} + 1} \cdot \frac{2}{e^{x \cdot -2} + 1}\right)\right) \cdot \left(1 + \frac{2}{e^{x \cdot -2} + 1}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))