Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 10.0

Time: 13.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\\ \mathbf{if}\;J \leq -3.733683180945843 \cdot 10^{-157}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(t_0 \cdot t_1\right)\\ \mathbf{elif}\;J \leq -7.086648613513012 \cdot 10^{-283}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.096025868775934 \cdot 10^{-206}:\\ \;\;\;\;-\mathsf{fma}\left(2, \frac{\left(J \cdot J\right) \cdot {\cos \left(K \cdot 0.5\right)}^{2}}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\left(J \cdot -2\right) \cdot t_0\right)\\ \end{array} \]

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 (hypot 1.0 (/ U (* t_0 (* J 2.0))))))
   (if (<= J -3.733683180945843e-157)
     (* (* J -2.0) (* t_0 t_1))
     (if (<= J -7.086648613513012e-283)
       U
       (if (<= J 9.096025868775934e-206)
         (- (fma 2.0 (/ (* (* J J) (pow (cos (* K 0.5)) 2.0)) U) U))
         (* t_1 (* (* J -2.0) t_0)))))))

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 = hypot(1.0, (U / (t_0 * (J * 2.0))));
	double tmp;
	if (J <= -3.733683180945843e-157) {
		tmp = (J * -2.0) * (t_0 * t_1);
	} else if (J <= -7.086648613513012e-283) {
		tmp = U;
	} else if (J <= 9.096025868775934e-206) {
		tmp = -fma(2.0, (((J * J) * pow(cos((K * 0.5)), 2.0)) / U), U);
	} else {
		tmp = t_1 * ((J * -2.0) * t_0);
	}
	return tmp;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 4 regimes

2. if J < -3.7336831809458427e-157

   1. Initial program 11.4

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

   2. Simplified 2.8

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

   3. Applied associate-*l*_binary64 2.9

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

   if -3.7336831809458427e-157 < J < -7.08664861351301211e-283

   1. Initial program 38.1

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

   2. Simplified 24.6

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

   3. Taylor expanded in U around -inf 35.3

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

   if -7.08664861351301211e-283 < J < 9.09602586877593381e-206

   1. Initial program 44.1

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

   2. Simplified 28.3

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

   3. Taylor expanded in J around 0 33.6

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

   4. Simplified 33.6

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

   if 9.09602586877593381e-206 < J

   1. Initial program 12.6

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

   2. Simplified 4.2

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

   3. Applied pow1_binary64 4.2

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

   4. Applied pow1_binary64 4.2

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

   5. Applied pow-prod-down_binary64 4.2

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

   6. Applied pow1_binary64 4.2

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

   7. Applied pow-prod-down_binary64 4.2

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

   8. Simplified 4.2

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{{\color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)}}^{1}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.733683180945843 \cdot 10^{-157}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)\right)\\ \mathbf{elif}\;J \leq -7.086648613513012 \cdot 10^{-283}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.096025868775934 \cdot 10^{-206}:\\ \;\;\;\;-\mathsf{fma}\left(2, \frac{\left(J \cdot J\right) \cdot {\cos \left(K \cdot 0.5\right)}^{2}}{U}, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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)))))