Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 13.2

Time: 10.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty \lor \neg \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2.729897108467623 \cdot 10^{+305}\right):\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{0.25}}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \end{array}\]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* -2.0 J) (cos (/ K 2.0)))
  (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))

↓

(FPCore (J K U)
 :precision binary64
 (if (or (<=
          (*
           (* (* -2.0 J) (cos (/ K 2.0)))
           (sqrt (+ 1.0 (pow (/ U (* (cos (/ K 2.0)) (* J 2.0))) 2.0))))
          (- INFINITY))
         (not
          (<=
           (*
            (* (* -2.0 J) (cos (/ K 2.0)))
            (sqrt (+ 1.0 (pow (/ U (* (cos (/ K 2.0)) (* J 2.0))) 2.0))))
           2.729897108467623e+305)))
   (*
    (* (* -2.0 J) (cos (/ K 2.0)))
    (* U (/ (sqrt 0.25) (* J (cos (* K 0.5))))))
   (*
    (* (* -2.0 J) (cos (/ K 2.0)))
    (sqrt (+ 1.0 (pow (/ U (* (cos (/ K 2.0)) (* J 2.0))) 2.0))))))

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 tmp;
	if (((((double) (((double) (((double) (-2.0 * J)) * ((double) cos((K / 2.0))))) * ((double) sqrt(((double) (1.0 + ((double) pow((U / ((double) (((double) cos((K / 2.0))) * ((double) (J * 2.0))))), 2.0)))))))) <= ((double) -(((double) INFINITY)))) || !(((double) (((double) (((double) (-2.0 * J)) * ((double) cos((K / 2.0))))) * ((double) sqrt(((double) (1.0 + ((double) pow((U / ((double) (((double) cos((K / 2.0))) * ((double) (J * 2.0))))), 2.0)))))))) <= 2.729897108467623e+305))) {
		tmp = ((double) (((double) (((double) (-2.0 * J)) * ((double) cos((K / 2.0))))) * ((double) (U * (((double) sqrt(0.25)) / ((double) (J * ((double) cos(((double) (K * 0.5)))))))))));
	} else {
		tmp = ((double) (((double) (((double) (-2.0 * J)) * ((double) cos((K / 2.0))))) * ((double) sqrt(((double) (1.0 + ((double) pow((U / ((double) (((double) cos((K / 2.0))) * ((double) (J * 2.0))))), 2.0))))))));
	}
	return tmp;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 2 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 or 2.72989710846762305e305 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0))))

   1. Initial program Error: 63.5 bits

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

   2. Taylor expanded around inf Error: 46.3 bits

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

   3. Simplified Error: 46.3 bits

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

   if -inf.0 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))) < 2.72989710846762305e305

   1. Initial program Error: 0.1 bits

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification Error: 13.2 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty \lor \neg \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2.729897108467623 \cdot 10^{+305}\right):\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{0.25}}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))))