Maksimov and Kolovsky, Equation (3)

Average Error: 17.6 → 8.4

Time: 12.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := t_0 \cdot \left(J \cdot -2\right)\\ \mathbf{if}\;U \leq 4.922700905671232 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(2 \cdot J\right)}\right) \cdot t_1\\ \mathbf{elif}\;U \leq 5.864597329753935 \cdot 10^{+238}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{t_0}}{2 \cdot J}\right)\\ \end{array} \]

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 (* t_0 (* J -2.0))))
   (if (<= U 4.922700905671232e+193)
     (* (hypot 1.0 (/ U (* t_0 (* 2.0 J)))) t_1)
     (if (<= U 5.864597329753935e+238)
       U
       (* t_1 (hypot 1.0 (/ (/ U t_0) (* 2.0 J))))))))

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 = t_0 * (J * -2.0);
	double tmp;
	if (U <= 4.922700905671232e+193) {
		tmp = hypot(1.0, (U / (t_0 * (2.0 * J)))) * t_1;
	} else if (U <= 5.864597329753935e+238) {
		tmp = U;
	} else {
		tmp = t_1 * hypot(1.0, ((U / t_0) / (2.0 * J)));
	}
	return tmp;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 3 regimes

2. if U < 4.92270090567123186e193

   1. Initial program 15.4

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

   2. Simplified 6.1

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

   3. Applied add-sqr-sqrt_binary64 6.3

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

   4. Applied associate-*r*_binary64 6.3

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

   5. Applied pow1_binary64 6.3

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

   6. Applied pow1_binary64 6.3

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

   7. Applied pow1_binary64 6.3

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

   8. Applied pow1_binary64 6.3

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

   9. Applied pow1_binary64 6.3

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

   10. Applied pow-prod-down_binary64 6.3

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

   11. Applied pow-prod-down_binary64 6.3

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

   12. Applied pow-prod-down_binary64 6.3

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

   13. Applied pow-prod-down_binary64 6.3

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

   14. Simplified 6.1

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

   if 4.92270090567123186e193 < U < 5.8645973297539346e238

   1. Initial program 36.0

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

   2. Simplified 20.8

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

   3. Taylor expanded in U around -inf 37.3

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

   if 5.8645973297539346e238 < U

   1. Initial program 42.9

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

   2. Simplified 27.8

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

   3. Applied associate-/r*_binary64 28.0

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 4.922700905671232 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{elif}\;U \leq 5.864597329753935 \cdot 10^{+238}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{\cos \left(\frac{K}{2}\right)}}{2 \cdot J}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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)))))