Logistic function from Lakshay Garg

Average Error: 29.4 → 0.2

Time: 4.5s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.168673542136596771:\\ \;\;\;\;1 \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\\ \mathbf{elif}\;-2 \cdot x \le 9.00792072330781855 \cdot 10^{-11}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - \sqrt{1}\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double temp;
	if (((-2.0 * x) <= -0.16867354213659677)) {
		temp = (1.0 * ((2.0 / (exp((-2.0 * x)) + 1.0)) - 1.0));
	} else {
		double temp_1;
		if (((-2.0 * x) <= 9.007920723307819e-11)) {
			temp_1 = (1.0 * fma(1.0, x, -fma(5.551115123125783e-17, pow(x, 4.0), (0.33333333333333337 * pow(x, 3.0)))));
		} else {
			temp_1 = (1.0 * ((sqrt((2.0 / (exp((-2.0 * x)) + 1.0))) + sqrt(1.0)) * (sqrt((2.0 / (exp((-2.0 * x)) + 1.0))) - sqrt(1.0))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if (* -2.0 x) < -0.16867354213659677

   1. Initial program 0.0

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

   3. Applied *-un-lft-identity 0.0

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

   4. Applied *-un-lft-identity 0.0

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

   5. Applied distribute-lft-out-- 0.0

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

   6. Simplified 0.0

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

   if -0.16867354213659677 < (* -2.0 x) < 9.007920723307819e-11

   1. Initial program 59.2

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

   3. Applied *-un-lft-identity 59.2

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

   4. Applied *-un-lft-identity 59.2

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

   5. Applied distribute-lft-out-- 59.2

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

   6. Simplified 59.2

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

   7. Taylor expanded around 0 0.1

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

   8. Simplified 0.1

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

   if 9.007920723307819e-11 < (* -2.0 x)

   1. Initial program 0.5

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

   3. Applied *-un-lft-identity 0.5

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

   4. Applied *-un-lft-identity 0.5

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

   5. Applied distribute-lft-out-- 0.5

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

   6. Simplified 0.5

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

   8. Applied add-sqr-sqrt 0.5

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

   9. Applied add-sqr-sqrt 0.5

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

   10. Applied difference-of-squares 0.5

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.168673542136596771:\\ \;\;\;\;1 \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\\ \mathbf{elif}\;-2 \cdot x \le 9.00792072330781855 \cdot 10^{-11}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - \sqrt{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))