Maksimov and Kolovsky, Equation (3)

Average Error: 17.8 → 8.1

Time: 14.1s

Precision: binary64

Math

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

↓

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

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 (/ U (* t_0 (* J 2.0))))
        (t_2 (* (* -2.0 J) t_0)))
   (if (<= (* t_2 (sqrt (+ 1.0 (pow t_1 2.0)))) (- INFINITY))
     (- U)
     (* t_2 (hypot 1.0 t_1)))))

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 = U / (t_0 * (J * 2.0));
	double t_2 = (-2.0 * J) * t_0;
	double tmp;
	if ((t_2 * sqrt(1.0 + pow(t_1, 2.0))) <= -((double) INFINITY)) {
		tmp = -U;
	} else {
		tmp = t_2 * hypot(1.0, t_1);
	}
	return tmp;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 2 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. Simplified 29.5

      \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)} \]

   3. Taylor expanded in J around 0 30.1

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

   4. Simplified 30.1

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

   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. Initial program 10.6

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

   2. Simplified 4.7

      \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.1

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

Reproduce

herbie shell --seed 2021225 
(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)))))