Logistic function from Lakshay Garg

Average Error: 29.8 → 0.0

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0008523613938262117:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3}} - 1\\ \mathbf{elif}\;x \leq 0.0007993552140358844:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} - 1 \cdot 1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -0.0008523613938262117)
   (- (cbrt (pow (/ 2.0 (+ 1.0 (pow (exp -2.0) x))) 3.0)) 1.0)
   (if (<= x 0.0007993552140358844)
     (-
      (* x 1.0)
      (* (pow x 3.0) (+ (* x 5.551115123125783e-17) 0.33333333333333337)))
     (/
      (- (pow (/ 2.0 (+ 1.0 (pow (exp -2.0) x))) 2.0) (* 1.0 1.0))
      (+ 1.0 (/ 2.0 (+ 1.0 (pow (exp -2.0) x))))))))

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 ((x <= -0.0008523613938262117)) {
		tmp = ((double) (((double) cbrt(((double) pow((2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))), 3.0)))) - 1.0));
	} else {
		double tmp_1;
		if ((x <= 0.0007993552140358844)) {
			tmp_1 = ((double) (((double) (x * 1.0)) - ((double) (((double) pow(x, 3.0)) * ((double) (((double) (x * 5.551115123125783e-17)) + 0.33333333333333337))))));
		} else {
			tmp_1 = (((double) (((double) pow((2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))), 2.0)) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + (2.0 / ((double) (1.0 + ((double) pow(((double) exp(-2.0)), x))))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if x < -8.5236139382621171e-4

   1. Initial program Error: 0.0 bits

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

   3. Applied add-cbrt-cube Error: 0.0 bits

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

   4. Applied add-cbrt-cube Error: 0.0 bits

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

   5. Applied cbrt-undiv Error: 0.0 bits

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

   6. Simplified Error: 0.0 bits

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

   if -8.5236139382621171e-4 < x < 7.993552140358844e-4

   1. Initial program Error: 59.1 bits

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

   2. Taylor expanded around 0 Error: 0.0 bits

      \[\leadsto \color{blue}{1 \cdot x - \left(5.551115123125783 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

   3. Simplified Error: 0.0 bits

      \[\leadsto \color{blue}{1 \cdot x - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)}\]

   if 7.993552140358844e-4 < x

   1. Initial program Error: 0.0 bits

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

   3. Applied add-cbrt-cube Error: 0.1 bits

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

   4. Applied add-cbrt-cube Error: 0.1 bits

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

   5. Applied cbrt-undiv Error: 0.1 bits

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

   6. Simplified Error: 0.1 bits

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

   8. Applied flip-- Error: 0.1 bits

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

   9. Simplified Error: 0.0 bits

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

   10. Simplified Error: 0.0 bits

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

4. Final simplification Error: 0.0 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0008523613938262117:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{3}} - 1\\ \mathbf{elif}\;x \leq 0.0007993552140358844:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right)}^{2} - 1 \cdot 1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\\ \end{array}\]

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))