Maksimov and Kolovsky, Equation (3)

Average Error: 17.8 → 13.3

Time: 11.3s

Precision: binary64

Math

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} = -inf.0:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{0.25}}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \le 2.9562275782773536 \cdot 10^{301}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(U \cdot \frac{\sqrt{0.25}}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\right)\right)\\ \end{array}\]

C

double code(double J, double K, double U) {
	return ((double) (((double) (((double) (-2.0 * J)) * ((double) cos(((double) (K / 2.0)))))) * ((double) sqrt(((double) (1.0 + ((double) pow(((double) (U / ((double) (((double) (2.0 * J)) * ((double) cos(((double) (K / 2.0)))))))), 2.0))))))));
}

↓

double code(double J, double K, double U) {
	double VAR;
	if ((((double) (((double) (((double) (-2.0 * J)) * ((double) cos(((double) (K / 2.0)))))) * ((double) sqrt(((double) (1.0 + ((double) pow(((double) (U / ((double) (((double) cos(((double) (K / 2.0)))) * ((double) (J * 2.0)))))), 2.0)))))))) <= -inf.0)) {
		VAR = ((double) (((double) (((double) (-2.0 * J)) * ((double) cos(((double) (K / 2.0)))))) * ((double) (U * ((double) (((double) sqrt(0.25)) / ((double) (J * ((double) cos(((double) (K * 0.5))))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (-2.0 * J)) * ((double) cos(((double) (K / 2.0)))))) * ((double) sqrt(((double) (1.0 + ((double) pow(((double) (U / ((double) (((double) cos(((double) (K / 2.0)))) * ((double) (J * 2.0)))))), 2.0)))))))) <= 2.9562275782773536e+301)) {
			VAR_1 = ((double) (((double) (((double) (-2.0 * J)) * ((double) cos(((double) (K / 2.0)))))) * ((double) sqrt(((double) (1.0 + ((double) pow(((double) (U / ((double) (((double) cos(((double) (K / 2.0)))) * ((double) (J * 2.0)))))), 2.0))))))));
		} else {
			VAR_1 = ((double) (-2.0 * ((double) (J * ((double) (((double) cos(((double) (K / 2.0)))) * ((double) (U * ((double) (((double) sqrt(0.25)) / ((double) (J * ((double) cos(((double) (K * 0.5))))))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 3 regimes

2. if (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))) < -inf.0

   1. Initial program 64.0

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

   2. Taylor expanded around inf 46.3

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}}\]

   3. Simplified 46.4

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(U \cdot \frac{\sqrt{0.25}}{J \cdot \cos \left(K \cdot 0.5\right)}\right)}\]

   if -inf.0 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))) < 2.9562275782773536e301

   1. Initial program 0.1

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

   if 2.9562275782773536e301 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0))))

   1. Initial program 60.6

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

   2. Simplified 60.5

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}^{2}}\right)\right)}\]

   3. Taylor expanded around inf 46.5

      \[\leadsto -2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}}\right)\right)\]

   4. Simplified 46.5

      \[\leadsto -2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(U \cdot \frac{\sqrt{0.25}}{J \cdot \cos \left(K \cdot 0.5\right)}\right)}\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 13.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} = -inf.0:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{0.25}}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \le 2.9562275782773536 \cdot 10^{301}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(U \cdot \frac{\sqrt{0.25}}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))