Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 13.7

Time: 12.5s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} = -inf.0:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \frac{U \cdot \sqrt{0.25}}{J \cdot \cos \left(0.5 \cdot K\right)}\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \le 1.7811757699890062 \cdot 10^{305}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{0.25}}{J \cdot \cos \left(0.5 \cdot K\right)}\\ \end{array}\]

C

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

↓

double code(double J, double K, double U) {
	double VAR;
	if ((((double) (((double) (((double) (-2.0 * J)) * ((double) cos((K / 2.0))))) * ((double) sqrt(((double) (1.0 + ((double) pow((U / ((double) (((double) (2.0 * J)) * ((double) cos((K / 2.0)))))), 2.0)))))))) <= -inf.0)) {
		VAR = ((double) (((double) (-2.0 * J)) * ((double) (((double) cos((K / 2.0))) * (((double) (U * ((double) sqrt(0.25)))) / ((double) (J * ((double) cos(((double) (0.5 * K)))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (-2.0 * J)) * ((double) cos((K / 2.0))))) * ((double) sqrt(((double) (1.0 + ((double) pow((U / ((double) (((double) (2.0 * J)) * ((double) cos((K / 2.0)))))), 2.0)))))))) <= 1.7811757699890062e+305)) {
			VAR_1 = ((double) (((double) (((double) (-2.0 * J)) * ((double) cos((K / 2.0))))) * ((double) sqrt(((double) (1.0 + ((double) pow((U / ((double) (((double) (2.0 * J)) * ((double) cos((K / 2.0)))))), 2.0))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (-2.0 * J)) * ((double) cos((K / 2.0))))) * (((double) (U * ((double) sqrt(0.25)))) / ((double) (J * ((double) cos(((double) (0.5 * K)))))))));
		}
		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. Using strategy rm

   3. Applied associate-*l* 64.0

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

   4. Taylor expanded around inf 46.7

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

   if -inf.0 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))) < 1.7811757699890062e305

   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 1.7811757699890062e305 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0))))

   1. Initial program 62.7

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

   2. Taylor expanded around inf 47.7

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

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} = -inf.0:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \frac{U \cdot \sqrt{0.25}}{J \cdot \cos \left(0.5 \cdot K\right)}\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \le 1.7811757699890062 \cdot 10^{305}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{0.25}}{J \cdot \cos \left(0.5 \cdot K\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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)))))