Maksimov and Kolovsky, Equation (3)

Average Error: 17.9 → 9.6

Time: 13.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot t_1\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \frac{{\cos \left(K \cdot 0.5\right)}^{2} \cdot \left(J \cdot J\right)}{U} - U\\ \mathbf{elif}\;t_2 \leq 6.6651908102113915 \cdot 10^{+295}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

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
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))
        (t_2 (* (* (* -2.0 J) t_0) t_1)))
   (if (<= t_2 (- INFINITY))
     (- (* -2.0 (/ (* (pow (cos (* K 0.5)) 2.0) (* J J)) U)) U)
     (if (<= t_2 6.6651908102113915e+295) (* (* -2.0 J) (* t_0 t_1)) U))))

C

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

↓

double code(double J, double K, double U) {
	double t_0 = cos(K / 2.0);
	double t_1 = sqrt(1.0 + pow((U / (t_0 * (J * 2.0))), 2.0));
	double t_2 = ((-2.0 * J) * t_0) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (-2.0 * ((pow(cos(K * 0.5), 2.0) * (J * J)) / U)) - U;
	} else if (t_2 <= 6.6651908102113915e+295) {
		tmp = (-2.0 * J) * (t_0 * t_1);
	} else {
		tmp = U;
	}
	return tmp;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < -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 flip3-+_binary64 64.0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{{1}^{3} + {\left({\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}\right)}^{3}}{1 \cdot 1 + \left({\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} - 1 \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}\right)}}} \]

   4. Applied sqrt-div_binary64 64.0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{{1}^{3} + {\left({\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}\right)}^{3}}}{\sqrt{1 \cdot 1 + \left({\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} - 1 \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}\right)}}} \]

   5. Applied associate-*r/_binary64 64.0

      \[\leadsto \color{blue}{\frac{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{{1}^{3} + {\left({\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}\right)}^{3}}}{\sqrt{1 \cdot 1 + \left({\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} - 1 \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}\right)}}} \]

   6. Simplified 64.0

      \[\leadsto \frac{\color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{6}} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right)}}{\sqrt{1 \cdot 1 + \left({\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} - 1 \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}\right)}} \]

   7. Taylor expanded around inf 31.5

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

   8. Simplified 31.5

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 6.6651908102113915e295

   1. Initial program 0.1

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

   3. Applied associate-*l*_binary64 0.2

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

   if 6.6651908102113915e295 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 58.8

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

   2. Taylor expanded around -inf 33.6

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-2 \cdot \frac{{\cos \left(K \cdot 0.5\right)}^{2} \cdot \left(J \cdot J\right)}{U} - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 6.6651908102113915 \cdot 10^{+295}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Reproduce

herbie shell --seed 2021205 
(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)))))