Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 8.4

Time: 19.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(J \cdot -2\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\\ \mathbf{if}\;J \leq -8.778931361479822 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 3.03198560535342 \cdot 10^{-237}:\\ \;\;\;\;-\mathsf{fma}\left(2, \frac{{\cos \left(K \cdot 0.5\right)}^{2} \cdot \left(J \cdot J\right)}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* (* (* J -2.0) t_0) (hypot 1.0 (/ U (* t_0 (* J 2.0)))))))
   (if (<= J -8.778931361479822e-290)
     t_1
     (if (<= J 3.03198560535342e-237)
       (- (fma 2.0 (/ (* (pow (cos (* K 0.5)) 2.0) (* J J)) U) U))
       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 = ((J * -2.0) * t_0) * hypot(1.0, (U / (t_0 * (J * 2.0))));
	double tmp;
	if (J <= -8.778931361479822e-290) {
		tmp = t_1;
	} else if (J <= 3.03198560535342e-237) {
		tmp = -fma(2.0, ((pow(cos(K * 0.5), 2.0) * (J * J)) / U), U);
	} else {
		tmp = 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 J < -8.778931361479822e-290 or 3.0319856053534199e-237 < J

   1. Initial program 16.0

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

   2. Simplified 6.7

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

   3. Applied add-cube-cbrt_binary64 7.2

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

   4. Applied associate-*r*_binary64 7.2

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

   5. Applied pow1_binary64 7.2

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

   6. Applied pow1_binary64 7.2

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

   7. Applied pow1_binary64 7.2

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

   8. Applied pow-prod-down_binary64 7.2

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

   9. Applied pow1_binary64 7.2

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

   10. Applied pow1_binary64 7.2

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

   11. Applied pow-prod-down_binary64 7.2

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

   12. Applied pow-prod-down_binary64 7.2

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

   13. Applied pow-prod-down_binary64 7.2

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

   14. Simplified 6.7

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

   if -8.778931361479822e-290 < J < 3.0319856053534199e-237

   1. Initial program 44.6

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

   2. Simplified 28.7

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

   3. Applied add-cube-cbrt_binary64 29.3

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

   4. Applied associate-*r*_binary64 29.3

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

   5. Taylor expanded in J around 0 31.1

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

   6. Simplified 31.1

      \[\leadsto \color{blue}{-\mathsf{fma}\left(2, \frac{{\cos \left(0.5 \cdot K\right)}^{2} \cdot \left(J \cdot J\right)}{U}, U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8.778931361479822 \cdot 10^{-290}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)\\ \mathbf{elif}\;J \leq 3.03198560535342 \cdot 10^{-237}:\\ \;\;\;\;-\mathsf{fma}\left(2, \frac{{\cos \left(K \cdot 0.5\right)}^{2} \cdot \left(J \cdot J\right)}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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)))))