Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 9.3

Time: 20.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 1.0935594133603016 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

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
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U)
     (if (<= t_1 1.0935594133603016e+307) 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 = ((-2.0 * J) * t_0) * sqrt(1.0 + pow((U / (t_0 * (J * 2.0))), 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 1.0935594133603016e+307) {
		tmp = 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. Taylor expanded in J around 0 32.8

      \[\leadsto \color{blue}{-1 \cdot U} \]

   3. Simplified 32.8

      \[\leadsto \color{blue}{-U} \]

   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)))) < 1.0935594133603016e307

   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 1.0935594133603016e307 < (*.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 63.6

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

   2. Taylor expanded in U around -inf 32.9

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 1.0935594133603016 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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)))))