Maksimov and Kolovsky, Equation (3)

Average Error: 18.0 → 17.3

Time: 21.1s

Precision: binary64

Cost: 28806

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;J \leq -2.1144643127566853 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{elif}\;J \leq -3.407987670226901 \cdot 10^{-287}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.129303200657122 \cdot 10^{-263}:\\ \;\;\;\;-2 \cdot \frac{\left(J \cdot J\right) \cdot {\cos \left(K \cdot 0.5\right)}^{2}}{U} - U\\ \mathbf{elif}\;J \leq 1.2804491241499647 \cdot 10^{-217}:\\ \;\;\;\;U + 2 \cdot \frac{\left(J \cdot J\right) \cdot {\cos \left(K \cdot 0.5\right)}^{2}}{U}\\ \mathbf{elif}\;J \leq 8.15590711171135 \cdot 10^{-182}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\right)\\ \mathbf{elif}\;J \leq 3.5351986706589427 \cdot 10^{-112}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \end{array}\]

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
 (if (<= J -2.1144643127566853e-111)
   (*
    (sqrt (+ 1.0 (pow (/ U (* (cos (/ K 2.0)) (* J 2.0))) 2.0)))
    (* (cos (/ K 2.0)) (* J -2.0)))
   (if (<= J -3.407987670226901e-287)
     U
     (if (<= J 5.129303200657122e-263)
       (- (* -2.0 (/ (* (* J J) (pow (cos (* K 0.5)) 2.0)) U)) U)
       (if (<= J 1.2804491241499647e-217)
         (+ U (* 2.0 (/ (* (* J J) (pow (cos (* K 0.5)) 2.0)) U)))
         (if (<= J 8.15590711171135e-182)
           (*
            (* J -2.0)
            (*
             (cos (/ K 2.0))
             (sqrt (+ 1.0 (pow (/ U (* (cos (/ K 2.0)) (* J 2.0))) 2.0)))))
           (if (<= J 3.5351986706589427e-112)
             U
             (*
              (sqrt (+ 1.0 (pow (/ U (* (cos (/ K 2.0)) (* J 2.0))) 2.0)))
              (* (cos (/ K 2.0)) (* J -2.0))))))))))

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 tmp;
	if (J <= -2.1144643127566853e-111) {
		tmp = sqrt(1.0 + pow((U / (cos(K / 2.0) * (J * 2.0))), 2.0)) * (cos(K / 2.0) * (J * -2.0));
	} else if (J <= -3.407987670226901e-287) {
		tmp = U;
	} else if (J <= 5.129303200657122e-263) {
		tmp = (-2.0 * (((J * J) * pow(cos(K * 0.5), 2.0)) / U)) - U;
	} else if (J <= 1.2804491241499647e-217) {
		tmp = U + (2.0 * (((J * J) * pow(cos(K * 0.5), 2.0)) / U));
	} else if (J <= 8.15590711171135e-182) {
		tmp = (J * -2.0) * (cos(K / 2.0) * sqrt(1.0 + pow((U / (cos(K / 2.0) * (J * 2.0))), 2.0)));
	} else if (J <= 3.5351986706589427e-112) {
		tmp = U;
	} else {
		tmp = sqrt(1.0 + pow((U / (cos(K / 2.0) * (J * 2.0))), 2.0)) * (cos(K / 2.0) * (J * -2.0));
	}
	return tmp;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Alternatives

Alternative 1
Error	18.9
Cost	99968

\[\sqrt[3]{\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot \left(\sqrt[3]{\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\]

Alternative 2
Error	18.4
Cost	72192

\[\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\left(\left(-2 \cdot J\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right)\right) \cdot \log \left(e^{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)\right)\]

Alternative 3
Error	41.3
Cost	66624

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

Alternative 4
Error	18.9
Cost	59904

\[\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\sqrt[3]{\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\right)\]

Alternative 5
Error	18.1
Cost	59712

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

Alternative 6
Error	18.1
Cost	59712

\[\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left|\sqrt[3]{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\right|\right) \cdot \sqrt{\sqrt[3]{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}}\]

Alternative 7
Error	18.1
Cost	59712

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

Alternative 8
Error	29.5
Cost	59456

\[\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\left(\left(-2 \cdot J\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right)\right) \cdot {\cos \left(\frac{K}{2}\right)}^{0.3333333333333333}\right)\]

Alternative 9
Error	18.4
Cost	59392

\[\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right)\right)\right)\]

Alternative 10
Error	31.2
Cost	52160

\[\left(\left(-2 \cdot J\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right)\right) \cdot \log \left(e^{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)\]

Alternative 11
Error	41.3
Cost	46592

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

Alternative 12
Error	47.8
Cost	39744

\[\sqrt[3]{{\left(\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)}^{3}}\]

Alternative 13
Error	47.8
Cost	39744

\[\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \sqrt[3]{{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)}^{3}}\]

Alternative 14
Error	31.2
Cost	39360

\[\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right)\right)\]

Alternative 15
Error	30.0
Cost	33344

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{0.5}{J}\right)}^{2} \cdot {\left(\frac{U}{\cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

Alternative 16
Error	30.0
Cost	27008

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \left(U \cdot U\right) \cdot {\left(\frac{\frac{0.5}{J}}{\cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

Alternative 17
Error	18.0
Cost	26880

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

Alternative 18
Error	18.0
Cost	26880

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

Alternative 19
Error	50.9
Cost	20800

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(\frac{U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot -0.5 - \frac{J \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\]

Alternative 20
Error	51.6
Cost	20800

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

Alternative 21
Error	38.8
Cost	20224

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

Alternative 22
Error	31.2
Cost	14016

\[-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right) - 0.25 \cdot \left(U \cdot \frac{U}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\]

Alternative 23
Error	32.3
Cost	14016

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\]

Alternative 24
Error	50.5
Cost	13888

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(\frac{U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot -0.5\right)\]

Alternative 25
Error	51.2
Cost	13888

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(0.5 \cdot \frac{U}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\]

Alternative 26
Error	48.0
Cost	13696

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

Alternative 27
Error	48.7
Cost	13696

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

Alternative 28
Error	44.3
Cost	7360

\[-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{U \cdot U}{J \cdot J}}\right)\]

Alternative 29
Error	30.9
Cost	6848

\[-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\]

Alternative 30
Error	30.9
Cost	6848

\[\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\]

Alternative 31
Error	47.2
Cost	128

\[-U\]

Alternative 32
Error	46.5
Cost	64

\[U\]

Alternative 33
Error	61.9
Cost	64

\[1\]

Alternative 34
Error	62.3
Cost	64

\[0\]

Alternative 35
Error	61.9
Cost	64

\[-1\]

Derivation

1. Split input into 5 regimes

2. if J < -2.1144643127566853e-111 or 3.53519867065894268e-112 < J

   1. Initial program 8.7

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

   2. Simplified 8.7

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

   if -2.1144643127566853e-111 < J < -3.407987670226901e-287 or 8.15590711171135e-182 < J < 3.53519867065894268e-112

   1. Initial program 35.3

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

   2. Taylor expanded around -inf 36.7

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

   3. Simplified 36.7

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

   if -3.407987670226901e-287 < J < 5.1293032006571219e-263

   1. Initial program 45.9

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

   2. Taylor expanded around 0 32.0

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

   3. Simplified 32.0

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

   4. Simplified 32.0

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

   if 5.1293032006571219e-263 < J < 1.2804491241499647e-217

   1. Initial program 39.8

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

   2. Taylor expanded around -inf 34.4

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

   3. Simplified 34.4

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

   4. Simplified 34.4

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

   if 1.2804491241499647e-217 < J < 8.15590711171135e-182

   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. Using strategy rm

   3. Applied associate-*l*_binary64_1011 36.0

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

   4. Simplified 36.0

      \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.1144643127566853 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{elif}\;J \leq -3.407987670226901 \cdot 10^{-287}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 5.129303200657122 \cdot 10^{-263}:\\ \;\;\;\;-2 \cdot \frac{\left(J \cdot J\right) \cdot {\cos \left(K \cdot 0.5\right)}^{2}}{U} - U\\ \mathbf{elif}\;J \leq 1.2804491241499647 \cdot 10^{-217}:\\ \;\;\;\;U + 2 \cdot \frac{\left(J \cdot J\right) \cdot {\cos \left(K \cdot 0.5\right)}^{2}}{U}\\ \mathbf{elif}\;J \leq 8.15590711171135 \cdot 10^{-182}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\right)\\ \mathbf{elif}\;J \leq 3.5351986706589427 \cdot 10^{-112}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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)))))