Logistic function from Lakshay Garg

Average Error: 29.8 → 0.1

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1.0337968322908504:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{2} - 1 \cdot 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 3.6353306986864113 \cdot 10^{-06}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 4.440892098500626 \cdot 10^{-16} + 0.3333333333333335\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}} - 1\\ \end{array}\]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -1.0337968322908504)
   (/
    (- (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 2.0) (* 1.0 1.0))
    (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) 3.6353306986864113e-06)
     (-
      (* x 1.0)
      (* (pow x 3.0) (+ (* x 4.440892098500626e-16) 0.3333333333333335)))
     (- (cbrt (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 3.0)) 1.0))))

C

double code(double x, double y) {
	return ((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) - 1.0));
}

↓

double code(double x, double y) {
	double tmp;
	if ((((double) (-2.0 * x)) <= -1.0337968322908504)) {
		tmp = (((double) (((double) pow((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))), 2.0)) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))))));
	} else {
		double tmp_1;
		if ((((double) (-2.0 * x)) <= 3.6353306986864113e-06)) {
			tmp_1 = ((double) (((double) (x * 1.0)) - ((double) (((double) pow(x, 3.0)) * ((double) (((double) (x * 4.440892098500626e-16)) + 0.3333333333333335))))));
		} else {
			tmp_1 = ((double) (((double) cbrt(((double) pow((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))), 3.0)))) - 1.0));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2.0 x) < -1.0337968322908504

   1. Initial program 0.0

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

   3. Applied add-cbrt-cube_binary64 0.0

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

   4. Applied add-cbrt-cube_binary64 0.0

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

   5. Applied cbrt-undiv_binary64 0.0

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

   6. Simplified 0.0

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

   8. Applied flip--_binary64 0.0

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

   9. Simplified 0.0

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

   10. Simplified 0.0

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

   if -1.0337968322908504 < (*.f64 -2.0 x) < 3.63533069868641128e-6

   1. Initial program 59.1

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

   2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{1 \cdot x - \left(4.440892098500626 \cdot 10^{-16} \cdot {x}^{4} + 0.3333333333333335 \cdot {x}^{3}\right)}\]

   3. Simplified 0.1

      \[\leadsto \color{blue}{1 \cdot x - {x}^{3} \cdot \left(x \cdot 4.440892098500626 \cdot 10^{-16} + 0.3333333333333335\right)}\]

   if 3.63533069868641128e-6 < (*.f64 -2.0 x)

   1. Initial program 0.1

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

   3. Applied add-cbrt-cube_binary64 0.1

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

   4. Applied add-cbrt-cube_binary64 0.1

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

   5. Applied cbrt-undiv_binary64 0.1

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

   6. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1.0337968322908504:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{2} - 1 \cdot 1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 3.6353306986864113 \cdot 10^{-06}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 4.440892098500626 \cdot 10^{-16} + 0.3333333333333335\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))