Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.5% → 73.5%

Time: 15.5s

Alternatives: 15

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \cos \left(K \cdot 0.5\right)\\ \left(t\_0 \cdot -2\right) \cdot \sqrt{1 + {\left(\frac{0.5 \cdot U}{t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (cos (* K 0.5)))))
   (* (* t_0 -2.0) (sqrt (+ 1.0 (pow (/ (* 0.5 U) t_0) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = J * cos((K * 0.5));
	return (t_0 * -2.0) * sqrt((1.0 + pow(((0.5 * U) / t_0), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = j * cos((k * 0.5d0))
    code = (t_0 * (-2.0d0)) * sqrt((1.0d0 + (((0.5d0 * u) / t_0) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = J * Math.cos((K * 0.5));
	return (t_0 * -2.0) * Math.sqrt((1.0 + Math.pow(((0.5 * U) / t_0), 2.0)));
}

Python

def code(J, K, U):
	t_0 = J * math.cos((K * 0.5))
	return (t_0 * -2.0) * math.sqrt((1.0 + math.pow(((0.5 * U) / t_0), 2.0)))

Julia

function code(J, K, U)
	t_0 = Float64(J * cos(Float64(K * 0.5)))
	return Float64(Float64(t_0 * -2.0) * sqrt(Float64(1.0 + (Float64(Float64(0.5 * U) / t_0) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = J * cos((K * 0.5));
	tmp = (t_0 * -2.0) * sqrt((1.0 + (((0.5 * U) / t_0) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * -2.0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[(0.5 * U), $MachinePrecision] / t$95$0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.0%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 80.0

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 80.0

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

4. Applied egg-rr 80.0%

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

5. Taylor expanded in U around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 80.0

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

7. Simplified 80.0%

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

8. Final simplification 80.0%

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

Alternative 2: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\mathsf{fma}\left(J, 0.5, J \cdot \left(0.5 \cdot \cos K\right)\right)} \cdot \frac{U}{J \cdot 4}} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* J (cos (* K 0.5))) -2.0)
  (sqrt (+ 1.0 (* (/ U (fma J 0.5 (* J (* 0.5 (cos K))))) (/ U (* J 4.0)))))))

C

double code(double J, double K, double U) {
	return ((J * cos((K * 0.5))) * -2.0) * sqrt((1.0 + ((U / fma(J, 0.5, (J * (0.5 * cos(K))))) * (U / (J * 4.0)))));
}

Julia

function code(J, K, U)
	return Float64(Float64(Float64(J * cos(Float64(K * 0.5))) * -2.0) * sqrt(Float64(1.0 + Float64(Float64(U / fma(J, 0.5, Float64(J * Float64(0.5 * cos(K))))) * Float64(U / Float64(J * 4.0))))))
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(U / N[(J * 0.5 + N[(J * N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U / N[(J * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 80.0

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 80.0

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

4. Applied egg-rr 80.0%

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

6. Applied egg-rr 79.9%

   \[\leadsto \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\mathsf{fma}\left(0.5, \cos \left(K \cdot 1\right), 0.5\right) \cdot J} \cdot \frac{U}{4 \cdot J}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\left(\frac{1}{2} \cdot \cos \color{blue}{\left(K \cdot 1\right)} + \frac{1}{2}\right) \cdot J} \cdot \frac{U}{4 \cdot J}} \]

   2. lift-cos.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\left(\frac{1}{2} \cdot \color{blue}{\cos \left(K \cdot 1\right)} + \frac{1}{2}\right) \cdot J} \cdot \frac{U}{4 \cdot J}} \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right)} \cdot J} \cdot \frac{U}{4 \cdot J}} \]

   4. *-commutative N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\color{blue}{J \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right)}} \cdot \frac{U}{4 \cdot J}} \]

   5. lift-fma.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{J \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \left(K \cdot 1\right) + \frac{1}{2}\right)}} \cdot \frac{U}{4 \cdot J}} \]

   6. +-commutative N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{J \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(K \cdot 1\right)\right)}} \cdot \frac{U}{4 \cdot J}} \]

   7. *-commutative N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{J \cdot \left(\frac{1}{2} + \color{blue}{\cos \left(K \cdot 1\right) \cdot \frac{1}{2}}\right)} \cdot \frac{U}{4 \cdot J}} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{J \cdot \left(\frac{1}{2} + \cos \color{blue}{\left(K \cdot 1\right)} \cdot \frac{1}{2}\right)} \cdot \frac{U}{4 \cdot J}} \]

   9. *-rgt-identity N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{J \cdot \left(\frac{1}{2} + \cos \color{blue}{K} \cdot \frac{1}{2}\right)} \cdot \frac{U}{4 \cdot J}} \]

   10. distribute-lft-in N/A

       \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\color{blue}{J \cdot \frac{1}{2} + J \cdot \left(\cos K \cdot \frac{1}{2}\right)}} \cdot \frac{U}{4 \cdot J}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\color{blue}{\mathsf{fma}\left(J, \frac{1}{2}, J \cdot \left(\cos K \cdot \frac{1}{2}\right)\right)}} \cdot \frac{U}{4 \cdot J}} \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\mathsf{fma}\left(J, \frac{1}{2}, \color{blue}{J \cdot \left(\cos K \cdot \frac{1}{2}\right)}\right)} \cdot \frac{U}{4 \cdot J}} \]

   13. *-commutative N/A

       \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\mathsf{fma}\left(J, \frac{1}{2}, J \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos K\right)}\right)} \cdot \frac{U}{4 \cdot J}} \]

   14. lower-*.f64 79.9

       \[\leadsto \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\mathsf{fma}\left(J, 0.5, J \cdot \color{blue}{\left(0.5 \cdot \cos K\right)}\right)} \cdot \frac{U}{4 \cdot J}} \]

8. Applied egg-rr 79.9%

   \[\leadsto \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\color{blue}{\mathsf{fma}\left(J, 0.5, J \cdot \left(0.5 \cdot \cos K\right)\right)}} \cdot \frac{U}{4 \cdot J}} \]

9. Final simplification 79.9%

   \[\leadsto \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\mathsf{fma}\left(J, 0.5, J \cdot \left(0.5 \cdot \cos K\right)\right)} \cdot \frac{U}{J \cdot 4}} \]
10. Add Preprocessing

Alternative 3: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := \frac{U}{\left(J \cdot -2\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right)}\\ \mathbf{if}\;J \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{0.25}{J}, 1\right)}\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;J \cdot \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot \left(J \cdot 4\right)\right)}, 1\right)} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* J -2.0) (cos (/ K 2.0))))
        (t_1
         (/
          U
          (*
           (* J -2.0)
           (fma K (* K (fma 0.0026041666666666665 (* K K) -0.125)) 1.0)))))
   (if (<= J 4.8e-158)
     (* t_0 (sqrt (fma (/ U J) (* U (/ 0.25 J)) 1.0)))
     (if (<= J 4.8e+113)
       (*
        J
        (*
         (sqrt (fma U (/ U (* (fma 0.5 (cos K) 0.5) (* J (* J 4.0)))) 1.0))
         (* (cos (* K 0.5)) -2.0)))
       (* t_0 (sqrt (fma t_1 t_1 1.0)))))))

C

double code(double J, double K, double U) {
	double t_0 = (J * -2.0) * cos((K / 2.0));
	double t_1 = U / ((J * -2.0) * fma(K, (K * fma(0.0026041666666666665, (K * K), -0.125)), 1.0));
	double tmp;
	if (J <= 4.8e-158) {
		tmp = t_0 * sqrt(fma((U / J), (U * (0.25 / J)), 1.0));
	} else if (J <= 4.8e+113) {
		tmp = J * (sqrt(fma(U, (U / (fma(0.5, cos(K), 0.5) * (J * (J * 4.0)))), 1.0)) * (cos((K * 0.5)) * -2.0));
	} else {
		tmp = t_0 * sqrt(fma(t_1, t_1, 1.0));
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(Float64(J * -2.0) * cos(Float64(K / 2.0)))
	t_1 = Float64(U / Float64(Float64(J * -2.0) * fma(K, Float64(K * fma(0.0026041666666666665, Float64(K * K), -0.125)), 1.0)))
	tmp = 0.0
	if (J <= 4.8e-158)
		tmp = Float64(t_0 * sqrt(fma(Float64(U / J), Float64(U * Float64(0.25 / J)), 1.0)));
	elseif (J <= 4.8e+113)
		tmp = Float64(J * Float64(sqrt(fma(U, Float64(U / Float64(fma(0.5, cos(K), 0.5) * Float64(J * Float64(J * 4.0)))), 1.0)) * Float64(cos(Float64(K * 0.5)) * -2.0)));
	else
		tmp = Float64(t_0 * sqrt(fma(t_1, t_1, 1.0)));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(J * -2.0), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U / N[(N[(J * -2.0), $MachinePrecision] * N[(K * N[(K * N[(0.0026041666666666665 * N[(K * K), $MachinePrecision] + -0.125), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 4.8e-158], N[(t$95$0 * N[Sqrt[N[(N[(U / J), $MachinePrecision] * N[(U * N[(0.25 / J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 4.8e+113], N[(J * N[(N[Sqrt[N[(U * N[(U / N[(N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision] * N[(J * N[(J * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if J < 4.80000000000000015e-158

   1. Initial program 75.5%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.2

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

   5. Simplified 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 65.2

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 65.2

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

   7. Applied egg-rr 65.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{1}{\frac{2 \cdot J}{U} \cdot \frac{2 \cdot J}{U}}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

   9. Applied egg-rr 65.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{\color{blue}{J \cdot 4}} \cdot \frac{U}{J} + 1} \]

       2. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{J \cdot 4}} \cdot \frac{U}{J} + 1} \]

       3. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J \cdot 4} \cdot \color{blue}{\frac{U}{J}} + 1} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{J} \cdot \frac{U}{J \cdot 4}} + 1} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{U}{\color{blue}{J \cdot 4}} + 1} \]

       6. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{U}{\color{blue}{4 \cdot J}} + 1} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{U}{\color{blue}{4 \cdot J}} + 1} \]

       8. lower-fma.f64 65.2

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, \frac{U}{4 \cdot J}, 1\right)}} \]

       9. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \color{blue}{\frac{U}{4 \cdot J}}, 1\right)} \]

       10. div-inv N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \color{blue}{U \cdot \frac{1}{4 \cdot J}}, 1\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \color{blue}{U \cdot \frac{1}{4 \cdot J}}, 1\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{1}{\color{blue}{4 \cdot J}}, 1\right)} \]

       13. associate-/r* N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \color{blue}{\frac{\frac{1}{4}}{J}}, 1\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{\color{blue}{\frac{1}{4}}}{J}, 1\right)} \]

       15. lower-/.f64 65.2

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \color{blue}{\frac{0.25}{J}}, 1\right)} \]

   11. Applied egg-rr 65.2%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{0.25}{J}, 1\right)}} \]

   if 4.80000000000000015e-158 < J < 4.79999999999999966e113

   1. Initial program 76.9%

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

   4. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(K \cdot 0.5\right)\right)\right) \cdot \left(\left(J \cdot 2\right) \cdot \left(J \cdot 2\right)\right)}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot J} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 76.8

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(K \cdot 0.5\right)\right)\right) \cdot \left(\left(J \cdot 2\right) \cdot \left(J \cdot 2\right)\right)}}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot J \]

   6. Applied egg-rr 76.8%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\mathsf{fma}\left(0.5, \cos \left(K \cdot 1\right), 0.5\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot J \]

   if 4.79999999999999966e113 < J

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}}\right)}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}}\right)}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right)}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\color{blue}{K \cdot \left(K \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)} + 1\right)}\right)}^{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right), 1\right)}}\right)}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, \color{blue}{K \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}, 1\right)}\right)}^{2}} \]

      6. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \color{blue}{\left(\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)}, 1\right)}\right)}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(\color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right), 1\right)}\right)}^{2}} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot \frac{1}{384} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right), 1\right)}\right)}^{2}} \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(\color{blue}{K \cdot \left(K \cdot \frac{1}{384}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right), 1\right)}\right)}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(K \cdot \left(K \cdot \frac{1}{384}\right) + \color{blue}{\frac{-1}{8}}\right), 1\right)}\right)}^{2}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)}\right)}^{2}} \]

      12. lower-*.f64 97.8

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, \color{blue}{K \cdot 0.0026041666666666665}, -0.125\right), 1\right)}\right)}^{2}} \]

   5. Simplified 97.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, K \cdot 0.0026041666666666665, -0.125\right), 1\right)}}\right)}^{2}} \]

   7. Applied egg-rr 97.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right) \cdot \left(J \cdot -2\right)}, \frac{U}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right) \cdot \left(J \cdot -2\right)}, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{0.25}{J}, 1\right)}\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;J \cdot \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot \left(J \cdot 4\right)\right)}, 1\right)} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\left(J \cdot -2\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right)}, \frac{U}{\left(J \cdot -2\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := \frac{U}{\left(J \cdot -2\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right)}\\ \mathbf{if}\;J \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{0.25}{J}, 1\right)}\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{\left(J \cdot \left(J \cdot 4\right)\right) \cdot \mathsf{fma}\left(\cos K, 0.5, 0.5\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* J -2.0) (cos (/ K 2.0))))
        (t_1
         (/
          U
          (*
           (* J -2.0)
           (fma K (* K (fma 0.0026041666666666665 (* K K) -0.125)) 1.0)))))
   (if (<= J 4.8e-158)
     (* t_0 (sqrt (fma (/ U J) (* U (/ 0.25 J)) 1.0)))
     (if (<= J 4.8e+113)
       (*
        (cos (* K 0.5))
        (*
         (* J -2.0)
         (sqrt (fma U (/ U (* (* J (* J 4.0)) (fma (cos K) 0.5 0.5))) 1.0))))
       (* t_0 (sqrt (fma t_1 t_1 1.0)))))))

C

double code(double J, double K, double U) {
	double t_0 = (J * -2.0) * cos((K / 2.0));
	double t_1 = U / ((J * -2.0) * fma(K, (K * fma(0.0026041666666666665, (K * K), -0.125)), 1.0));
	double tmp;
	if (J <= 4.8e-158) {
		tmp = t_0 * sqrt(fma((U / J), (U * (0.25 / J)), 1.0));
	} else if (J <= 4.8e+113) {
		tmp = cos((K * 0.5)) * ((J * -2.0) * sqrt(fma(U, (U / ((J * (J * 4.0)) * fma(cos(K), 0.5, 0.5))), 1.0)));
	} else {
		tmp = t_0 * sqrt(fma(t_1, t_1, 1.0));
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(Float64(J * -2.0) * cos(Float64(K / 2.0)))
	t_1 = Float64(U / Float64(Float64(J * -2.0) * fma(K, Float64(K * fma(0.0026041666666666665, Float64(K * K), -0.125)), 1.0)))
	tmp = 0.0
	if (J <= 4.8e-158)
		tmp = Float64(t_0 * sqrt(fma(Float64(U / J), Float64(U * Float64(0.25 / J)), 1.0)));
	elseif (J <= 4.8e+113)
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(Float64(J * -2.0) * sqrt(fma(U, Float64(U / Float64(Float64(J * Float64(J * 4.0)) * fma(cos(K), 0.5, 0.5))), 1.0))));
	else
		tmp = Float64(t_0 * sqrt(fma(t_1, t_1, 1.0)));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(J * -2.0), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U / N[(N[(J * -2.0), $MachinePrecision] * N[(K * N[(K * N[(0.0026041666666666665 * N[(K * K), $MachinePrecision] + -0.125), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 4.8e-158], N[(t$95$0 * N[Sqrt[N[(N[(U / J), $MachinePrecision] * N[(U * N[(0.25 / J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 4.8e+113], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(U * N[(U / N[(N[(J * N[(J * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if J < 4.80000000000000015e-158

   1. Initial program 75.5%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.2

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

   5. Simplified 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 65.2

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 65.2

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

   7. Applied egg-rr 65.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{1}{\frac{2 \cdot J}{U} \cdot \frac{2 \cdot J}{U}}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

   9. Applied egg-rr 65.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{\color{blue}{J \cdot 4}} \cdot \frac{U}{J} + 1} \]

       2. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{J \cdot 4}} \cdot \frac{U}{J} + 1} \]

       3. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J \cdot 4} \cdot \color{blue}{\frac{U}{J}} + 1} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{J} \cdot \frac{U}{J \cdot 4}} + 1} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{U}{\color{blue}{J \cdot 4}} + 1} \]

       6. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{U}{\color{blue}{4 \cdot J}} + 1} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{U}{\color{blue}{4 \cdot J}} + 1} \]

       8. lower-fma.f64 65.2

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, \frac{U}{4 \cdot J}, 1\right)}} \]

       9. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \color{blue}{\frac{U}{4 \cdot J}}, 1\right)} \]

       10. div-inv N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \color{blue}{U \cdot \frac{1}{4 \cdot J}}, 1\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \color{blue}{U \cdot \frac{1}{4 \cdot J}}, 1\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{1}{\color{blue}{4 \cdot J}}, 1\right)} \]

       13. associate-/r* N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \color{blue}{\frac{\frac{1}{4}}{J}}, 1\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{\color{blue}{\frac{1}{4}}}{J}, 1\right)} \]

       15. lower-/.f64 65.2

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \color{blue}{\frac{0.25}{J}}, 1\right)} \]

   11. Applied egg-rr 65.2%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{0.25}{J}, 1\right)}} \]

   if 4.80000000000000015e-158 < J < 4.79999999999999966e113

   1. Initial program 76.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 76.9

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 76.9

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

   4. Applied egg-rr 76.9%

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

   6. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(0.5, \cos \left(K \cdot 1\right), 0.5\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\frac{1}{2} \cdot \cos \color{blue}{\left(K \cdot 1\right)} + \frac{1}{2}\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\frac{1}{2} \cdot \color{blue}{\cos \left(K \cdot 1\right)} + \frac{1}{2}\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\color{blue}{\cos \left(K \cdot 1\right) \cdot \frac{1}{2}} + \frac{1}{2}\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      4. lower-fma.f64 76.8

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 1\right), 0.5, 0.5\right)} \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(\cos \color{blue}{\left(K \cdot 1\right)}, \frac{1}{2}, \frac{1}{2}\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      6. *-rgt-identity 76.8

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(\cos \color{blue}{K}, 0.5, 0.5\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right) \]

   8. Applied egg-rr 76.8%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)} \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right) \]

   if 4.79999999999999966e113 < J

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}}\right)}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}}\right)}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right)}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\color{blue}{K \cdot \left(K \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)} + 1\right)}\right)}^{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right), 1\right)}}\right)}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, \color{blue}{K \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}, 1\right)}\right)}^{2}} \]

      6. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \color{blue}{\left(\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)}, 1\right)}\right)}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(\color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right), 1\right)}\right)}^{2}} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot \frac{1}{384} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right), 1\right)}\right)}^{2}} \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(\color{blue}{K \cdot \left(K \cdot \frac{1}{384}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right), 1\right)}\right)}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(K \cdot \left(K \cdot \frac{1}{384}\right) + \color{blue}{\frac{-1}{8}}\right), 1\right)}\right)}^{2}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)}\right)}^{2}} \]

      12. lower-*.f64 97.8

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, \color{blue}{K \cdot 0.0026041666666666665}, -0.125\right), 1\right)}\right)}^{2}} \]

   5. Simplified 97.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, K \cdot 0.0026041666666666665, -0.125\right), 1\right)}}\right)}^{2}} \]

   7. Applied egg-rr 97.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right) \cdot \left(J \cdot -2\right)}, \frac{U}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right) \cdot \left(J \cdot -2\right)}, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{0.25}{J}, 1\right)}\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{\left(J \cdot \left(J \cdot 4\right)\right) \cdot \mathsf{fma}\left(\cos K, 0.5, 0.5\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\left(J \cdot -2\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right)}, \frac{U}{\left(J \cdot -2\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_1 := \frac{U}{\left(J \cdot -2\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right)}\\ \mathbf{if}\;J \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{0.25}{J}, 1\right)}\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{J \cdot \left(\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot 4\right)\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* J -2.0) (cos (/ K 2.0))))
        (t_1
         (/
          U
          (*
           (* J -2.0)
           (fma K (* K (fma 0.0026041666666666665 (* K K) -0.125)) 1.0)))))
   (if (<= J 4.8e-158)
     (* t_0 (sqrt (fma (/ U J) (* U (/ 0.25 J)) 1.0)))
     (if (<= J 4.8e+113)
       (*
        (* J -2.0)
        (*
         (cos (* K 0.5))
         (sqrt (fma U (/ U (* J (* (fma 0.5 (cos K) 0.5) (* J 4.0)))) 1.0))))
       (* t_0 (sqrt (fma t_1 t_1 1.0)))))))

C

double code(double J, double K, double U) {
	double t_0 = (J * -2.0) * cos((K / 2.0));
	double t_1 = U / ((J * -2.0) * fma(K, (K * fma(0.0026041666666666665, (K * K), -0.125)), 1.0));
	double tmp;
	if (J <= 4.8e-158) {
		tmp = t_0 * sqrt(fma((U / J), (U * (0.25 / J)), 1.0));
	} else if (J <= 4.8e+113) {
		tmp = (J * -2.0) * (cos((K * 0.5)) * sqrt(fma(U, (U / (J * (fma(0.5, cos(K), 0.5) * (J * 4.0)))), 1.0)));
	} else {
		tmp = t_0 * sqrt(fma(t_1, t_1, 1.0));
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(Float64(J * -2.0) * cos(Float64(K / 2.0)))
	t_1 = Float64(U / Float64(Float64(J * -2.0) * fma(K, Float64(K * fma(0.0026041666666666665, Float64(K * K), -0.125)), 1.0)))
	tmp = 0.0
	if (J <= 4.8e-158)
		tmp = Float64(t_0 * sqrt(fma(Float64(U / J), Float64(U * Float64(0.25 / J)), 1.0)));
	elseif (J <= 4.8e+113)
		tmp = Float64(Float64(J * -2.0) * Float64(cos(Float64(K * 0.5)) * sqrt(fma(U, Float64(U / Float64(J * Float64(fma(0.5, cos(K), 0.5) * Float64(J * 4.0)))), 1.0))));
	else
		tmp = Float64(t_0 * sqrt(fma(t_1, t_1, 1.0)));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(J * -2.0), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U / N[(N[(J * -2.0), $MachinePrecision] * N[(K * N[(K * N[(0.0026041666666666665 * N[(K * K), $MachinePrecision] + -0.125), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 4.8e-158], N[(t$95$0 * N[Sqrt[N[(N[(U / J), $MachinePrecision] * N[(U * N[(0.25 / J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 4.8e+113], N[(N[(J * -2.0), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(U / N[(J * N[(N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision] * N[(J * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if J < 4.80000000000000015e-158

   1. Initial program 75.5%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.2

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

   5. Simplified 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 65.2

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 65.2

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

   7. Applied egg-rr 65.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{1}{\frac{2 \cdot J}{U} \cdot \frac{2 \cdot J}{U}}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

   9. Applied egg-rr 65.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{\color{blue}{J \cdot 4}} \cdot \frac{U}{J} + 1} \]

       2. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{J \cdot 4}} \cdot \frac{U}{J} + 1} \]

       3. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J \cdot 4} \cdot \color{blue}{\frac{U}{J}} + 1} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{J} \cdot \frac{U}{J \cdot 4}} + 1} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{U}{\color{blue}{J \cdot 4}} + 1} \]

       6. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{U}{\color{blue}{4 \cdot J}} + 1} \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{U}{\color{blue}{4 \cdot J}} + 1} \]

       8. lower-fma.f64 65.2

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, \frac{U}{4 \cdot J}, 1\right)}} \]

       9. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \color{blue}{\frac{U}{4 \cdot J}}, 1\right)} \]

       10. div-inv N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \color{blue}{U \cdot \frac{1}{4 \cdot J}}, 1\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, \color{blue}{U \cdot \frac{1}{4 \cdot J}}, 1\right)} \]

       12. lift-*.f64 N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{1}{\color{blue}{4 \cdot J}}, 1\right)} \]

       13. associate-/r* N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \color{blue}{\frac{\frac{1}{4}}{J}}, 1\right)} \]

       14. metadata-eval N/A

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{\color{blue}{\frac{1}{4}}}{J}, 1\right)} \]

       15. lower-/.f64 65.2

           \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \color{blue}{\frac{0.25}{J}}, 1\right)} \]

   11. Applied egg-rr 65.2%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{0.25}{J}, 1\right)}} \]

   if 4.80000000000000015e-158 < J < 4.79999999999999966e113

   1. Initial program 76.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 76.9

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 76.9

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

   4. Applied egg-rr 76.9%

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

   6. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(0.5, \cos \left(K \cdot 1\right), 0.5\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right)} \]

   8. Applied egg-rr 76.7%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(U, \frac{U}{J \cdot \left(\left(J \cdot 4\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)\right)}, 1\right)} \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot -2\right)} \]

   if 4.79999999999999966e113 < J

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}}\right)}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}}\right)}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right)}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\color{blue}{K \cdot \left(K \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)} + 1\right)}\right)}^{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right), 1\right)}}\right)}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, \color{blue}{K \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}, 1\right)}\right)}^{2}} \]

      6. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \color{blue}{\left(\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)}, 1\right)}\right)}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(\color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right), 1\right)}\right)}^{2}} \]

      8. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot \frac{1}{384} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right), 1\right)}\right)}^{2}} \]

      9. associate-*l* N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(\color{blue}{K \cdot \left(K \cdot \frac{1}{384}\right)} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right), 1\right)}\right)}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \left(K \cdot \left(K \cdot \frac{1}{384}\right) + \color{blue}{\frac{-1}{8}}\right), 1\right)}\right)}^{2}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)}\right)}^{2}} \]

      12. lower-*.f64 97.8

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, \color{blue}{K \cdot 0.0026041666666666665}, -0.125\right), 1\right)}\right)}^{2}} \]

   5. Simplified 97.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K, K \cdot 0.0026041666666666665, -0.125\right), 1\right)}}\right)}^{2}} \]

   7. Applied egg-rr 97.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right) \cdot \left(J \cdot -2\right)}, \frac{U}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right) \cdot \left(J \cdot -2\right)}, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, U \cdot \frac{0.25}{J}, 1\right)}\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{+113}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{J \cdot \left(\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot 4\right)\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\left(J \cdot -2\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right)}, \frac{U}{\left(J \cdot -2\right) \cdot \mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(0.0026041666666666665, K \cdot K, -0.125\right), 1\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq 1.02 \cdot 10^{-238}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{\frac{U}{J \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}}{J \cdot 4}, 1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= J 1.02e-238)
   (* (* J -2.0) (sqrt (fma (/ U (* J 4.0)) (/ U J) 1.0)))
   (*
    (cos (* K 0.5))
    (*
     (* J -2.0)
     (sqrt (fma U (/ (/ U (* J (fma 0.5 (cos K) 0.5))) (* J 4.0)) 1.0))))))

C

double code(double J, double K, double U) {
	double tmp;
	if (J <= 1.02e-238) {
		tmp = (J * -2.0) * sqrt(fma((U / (J * 4.0)), (U / J), 1.0));
	} else {
		tmp = cos((K * 0.5)) * ((J * -2.0) * sqrt(fma(U, ((U / (J * fma(0.5, cos(K), 0.5))) / (J * 4.0)), 1.0)));
	}
	return tmp;
}

Julia

function code(J, K, U)
	tmp = 0.0
	if (J <= 1.02e-238)
		tmp = Float64(Float64(J * -2.0) * sqrt(fma(Float64(U / Float64(J * 4.0)), Float64(U / J), 1.0)));
	else
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(Float64(J * -2.0) * sqrt(fma(U, Float64(Float64(U / Float64(J * fma(0.5, cos(K), 0.5))) / Float64(J * 4.0)), 1.0))));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := If[LessEqual[J, 1.02e-238], N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(U / N[(J * 4.0), $MachinePrecision]), $MachinePrecision] * N[(U / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(U * N[(N[(U / N[(J * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(J * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.01999999999999992e-238

   1. Initial program 78.3%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.3

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

   5. Simplified 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 68.2

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 68.2

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

   7. Applied egg-rr 68.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{1}{\frac{2 \cdot J}{U} \cdot \frac{2 \cdot J}{U}}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

   9. Applied egg-rr 68.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}} \]

   10. Taylor expanded in K around 0

       \[\leadsto \color{blue}{\left(-2 \cdot J\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

       2. lower-*.f64 45.1

          \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

   12. Simplified 45.1%

       \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

   if 1.01999999999999992e-238 < J

   1. Initial program 81.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 81.9

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 81.9

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

   4. Applied egg-rr 81.9%

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

   6. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(0.5, \cos \left(K \cdot 1\right), 0.5\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\frac{1}{2} \cdot \cos \color{blue}{\left(K \cdot 1\right)} + \frac{1}{2}\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\frac{1}{2} \cdot \color{blue}{\cos \left(K \cdot 1\right)} + \frac{1}{2}\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\color{blue}{\cos \left(K \cdot 1\right) \cdot \frac{1}{2}} + \frac{1}{2}\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      4. lower-fma.f64 71.4

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\mathsf{fma}\left(\cos \left(K \cdot 1\right), 0.5, 0.5\right)} \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(\cos \color{blue}{\left(K \cdot 1\right)}, \frac{1}{2}, \frac{1}{2}\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      6. *-rgt-identity 71.4

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(\cos \color{blue}{K}, 0.5, 0.5\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right) \]

   8. Applied egg-rr 71.4%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)} \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right) \]
   9. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\color{blue}{\cos K} \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)} \cdot \left(J \cdot \left(4 \cdot J\right)\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right) \cdot \left(J \cdot \color{blue}{\left(4 \cdot J\right)}\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right) \cdot \color{blue}{\left(J \cdot \left(4 \cdot J\right)\right)}}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right) \cdot \color{blue}{\left(J \cdot \left(4 \cdot J\right)\right)}}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\left(\mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right) \cdot J\right) \cdot \left(4 \cdot J\right)}}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\color{blue}{\left(\cos K \cdot \frac{1}{2} + \frac{1}{2}\right)} \cdot J\right) \cdot \left(4 \cdot J\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\left(\color{blue}{\frac{1}{2} \cdot \cos K} + \frac{1}{2}\right) \cdot J\right) \cdot \left(4 \cdot J\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos K, \frac{1}{2}\right)} \cdot J\right) \cdot \left(4 \cdot J\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(K \cdot 1\right)}, \frac{1}{2}\right) \cdot J\right) \cdot \left(4 \cdot J\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(\mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(K \cdot 1\right)}, \frac{1}{2}\right) \cdot J\right) \cdot \left(4 \cdot J\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J\right)} \cdot \left(4 \cdot J\right)}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      13. associate-/r* N/A

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \color{blue}{\frac{\frac{U}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J}}{4 \cdot J}}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{\color{blue}{\frac{U}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J}}}{4 \cdot J}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot \frac{1}{2}\right) \]

      15. lower-/.f64 81.0

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \color{blue}{\frac{\frac{U}{\mathsf{fma}\left(0.5, \cos \left(K \cdot 1\right), 0.5\right) \cdot J}}{4 \cdot J}}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right) \]

   10. Applied egg-rr 81.0%

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \color{blue}{\frac{\frac{U}{J \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}}{J \cdot 4}}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.02 \cdot 10^{-238}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{\frac{U}{J \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}}{J \cdot 4}, 1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{J \cdot 4} \cdot \frac{U}{J \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* J (cos (* K 0.5))) -2.0)
  (sqrt (+ 1.0 (* (/ U (* J 4.0)) (/ U (* J (fma 0.5 (cos K) 0.5))))))))

C

double code(double J, double K, double U) {
	return ((J * cos((K * 0.5))) * -2.0) * sqrt((1.0 + ((U / (J * 4.0)) * (U / (J * fma(0.5, cos(K), 0.5))))));
}

Julia

function code(J, K, U)
	return Float64(Float64(Float64(J * cos(Float64(K * 0.5))) * -2.0) * sqrt(Float64(1.0 + Float64(Float64(U / Float64(J * 4.0)) * Float64(U / Float64(J * fma(0.5, cos(K), 0.5)))))))
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(U / N[(J * 4.0), $MachinePrecision]), $MachinePrecision] * N[(U / N[(J * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 80.0

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 80.0

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

4. Applied egg-rr 80.0%

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

6. Applied egg-rr 79.9%

   \[\leadsto \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\mathsf{fma}\left(0.5, \cos \left(K \cdot 1\right), 0.5\right) \cdot J} \cdot \frac{U}{4 \cdot J}}} \]

7. Final simplification 79.9%

   \[\leadsto \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{J \cdot 4} \cdot \frac{U}{J \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}} \]
8. Add Preprocessing

Alternative 8: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, U \cdot \frac{0.25}{J}, 1\right)} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* J (cos (* K 0.5))) -2.0)
  (sqrt (fma (/ U (* J (fma 0.5 (cos K) 0.5))) (* U (/ 0.25 J)) 1.0))))

C

double code(double J, double K, double U) {
	return ((J * cos((K * 0.5))) * -2.0) * sqrt(fma((U / (J * fma(0.5, cos(K), 0.5))), (U * (0.25 / J)), 1.0));
}

Julia

function code(J, K, U)
	return Float64(Float64(Float64(J * cos(Float64(K * 0.5))) * -2.0) * sqrt(fma(Float64(U / Float64(J * fma(0.5, cos(K), 0.5))), Float64(U * Float64(0.25 / J)), 1.0)))
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(U / N[(J * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U * N[(0.25 / J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 80.0

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 80.0

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

4. Applied egg-rr 80.0%

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

6. Applied egg-rr 79.9%

   \[\leadsto \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\mathsf{fma}\left(0.5, \cos \left(K \cdot 1\right), 0.5\right) \cdot J} \cdot \frac{U}{4 \cdot J}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\left(\frac{1}{2} \cdot \cos \color{blue}{\left(K \cdot 1\right)} + \frac{1}{2}\right) \cdot J} \cdot \frac{U}{4 \cdot J}} \]

   2. lift-cos.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\left(\frac{1}{2} \cdot \color{blue}{\cos \left(K \cdot 1\right)} + \frac{1}{2}\right) \cdot J} \cdot \frac{U}{4 \cdot J}} \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right)} \cdot J} \cdot \frac{U}{4 \cdot J}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J}} \cdot \frac{U}{4 \cdot J}} \]

   5. lift-/.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J}} \cdot \frac{U}{4 \cdot J}} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J} \cdot \frac{U}{\color{blue}{4 \cdot J}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \frac{U}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J} \cdot \color{blue}{\frac{U}{4 \cdot J}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J} \cdot \frac{U}{4 \cdot J}}} \]

   9. +-commutative N/A

      \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{\color{blue}{\frac{U}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J} \cdot \frac{U}{4 \cdot J} + 1}} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(J \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot -2\right) \cdot \sqrt{\color{blue}{\frac{U}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(K \cdot 1\right), \frac{1}{2}\right) \cdot J} \cdot \frac{U}{4 \cdot J}} + 1} \]

   11. lower-fma.f64 79.9

       \[\leadsto \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(0.5, \cos \left(K \cdot 1\right), 0.5\right) \cdot J}, \frac{U}{4 \cdot J}, 1\right)}} \]

8. Applied egg-rr 79.8%

   \[\leadsto \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, U \cdot \frac{0.25}{J}, 1\right)}} \]
9. Add Preprocessing

Alternative 9: 62.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.999:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.999)
   (* (cos (* K 0.5)) (* J -2.0))
   (* (* J -2.0) (sqrt (fma (/ U (* J 4.0)) (/ U J) 1.0)))))

C

double code(double J, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.999) {
		tmp = cos((K * 0.5)) * (J * -2.0);
	} else {
		tmp = (J * -2.0) * sqrt(fma((U / (J * 4.0)), (U / J), 1.0));
	}
	return tmp;
}

Julia

function code(J, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.999)
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(J * -2.0));
	else
		tmp = Float64(Float64(J * -2.0) * sqrt(fma(Float64(U / Float64(J * 4.0)), Float64(U / J), 1.0)));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.999], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(U / N[(J * 4.0), $MachinePrecision]), $MachinePrecision] * N[(U / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.998999999999999999

   1. Initial program 83.7%

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

   3. Taylor expanded in J around inf

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 60.5

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

   5. Simplified 60.5%

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

   if 0.998999999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 76.6%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.7

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

   5. Simplified 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 75.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 75.7

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

   7. Applied egg-rr 75.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{1}{\frac{2 \cdot J}{U} \cdot \frac{2 \cdot J}{U}}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

   9. Applied egg-rr 75.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}} \]

   10. Taylor expanded in K around 0

       \[\leadsto \color{blue}{\left(-2 \cdot J\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

       2. lower-*.f64 76.2

          \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

   12. Simplified 76.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.999:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-J\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.04)
   (* (cos (* K 0.5)) (- J))
   (* (* J -2.0) (sqrt (fma (/ U (* J 4.0)) (/ U J) 1.0)))))

C

double code(double J, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.04) {
		tmp = cos((K * 0.5)) * -J;
	} else {
		tmp = (J * -2.0) * sqrt(fma((U / (J * 4.0)), (U / J), 1.0));
	}
	return tmp;
}

Julia

function code(J, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.04)
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(-J));
	else
		tmp = Float64(Float64(J * -2.0) * sqrt(fma(Float64(U / Float64(J * 4.0)), Float64(U / J), 1.0)));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.04], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * (-J)), $MachinePrecision], N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(U / N[(J * 4.0), $MachinePrecision]), $MachinePrecision] * N[(U / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0400000000000000008

   1. Initial program 85.4%

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

   3. Taylor expanded in J around inf

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 61.4

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

   5. Simplified 61.4%

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

   6. Taylor expanded in J around -inf

      \[\leadsto \cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(-1 \cdot J\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 13.8

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

   8. Simplified 13.8%

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

   if 0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 77.9%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.2

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

   5. Simplified 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 72.2

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 72.2

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

   7. Applied egg-rr 72.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{1}{\frac{2 \cdot J}{U} \cdot \frac{2 \cdot J}{U}}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

   9. Applied egg-rr 72.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}} \]

   10. Taylor expanded in K around 0

       \[\leadsto \color{blue}{\left(-2 \cdot J\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

       2. lower-*.f64 65.1

          \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

   12. Simplified 65.1%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.04:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-J\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(-K\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (* (* J -2.0) (cos (- K)))
   (* (* J -2.0) (sqrt (fma (/ U (* J 4.0)) (/ U J) 1.0)))))

C

double code(double J, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.05) {
		tmp = (J * -2.0) * cos(-K);
	} else {
		tmp = (J * -2.0) * sqrt(fma((U / (J * 4.0)), (U / J), 1.0));
	}
	return tmp;
}

Julia

function code(J, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.05)
		tmp = Float64(Float64(J * -2.0) * cos(Float64(-K)));
	else
		tmp = Float64(Float64(J * -2.0) * sqrt(fma(Float64(U / Float64(J * 4.0)), Float64(U / J), 1.0)));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.05], N[(N[(J * -2.0), $MachinePrecision] * N[Cos[(-K)], $MachinePrecision]), $MachinePrecision], N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(U / N[(J * 4.0), $MachinePrecision]), $MachinePrecision] * N[(U / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.050000000000000003

   1. Initial program 85.2%

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

   3. Taylor expanded in J around inf

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 60.9

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

   5. Simplified 60.9%

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

   6. Taylor expanded in K around -inf

      \[\leadsto \cos \color{blue}{\left(-1 \cdot K\right)} \cdot \left(J \cdot -2\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \cos \color{blue}{\left(\mathsf{neg}\left(K\right)\right)} \cdot \left(J \cdot -2\right) \]

      2. lower-neg.f64 7.3

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

   8. Simplified 7.3%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 78.0%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.3

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

   5. Simplified 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 72.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 72.3

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

   7. Applied egg-rr 72.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{1}{\frac{2 \cdot J}{U} \cdot \frac{2 \cdot J}{U}}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

   9. Applied egg-rr 72.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}} \]

   10. Taylor expanded in K around 0

       \[\leadsto \color{blue}{\left(-2 \cdot J\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

       2. lower-*.f64 64.8

          \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

   12. Simplified 64.8%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(-K\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot -2\right)\right) \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (* (sqrt (fma (/ U (* J 4.0)) (/ U J) 1.0)) (* J (* (cos (* K 0.5)) -2.0))))

C

double code(double J, double K, double U) {
	return sqrt(fma((U / (J * 4.0)), (U / J), 1.0)) * (J * (cos((K * 0.5)) * -2.0));
}

Julia

function code(J, K, U)
	return Float64(sqrt(fma(Float64(U / Float64(J * 4.0)), Float64(U / J), 1.0)) * Float64(J * Float64(cos(Float64(K * 0.5)) * -2.0)))
end

Wolfram

code[J_, K_, U_] := N[(N[Sqrt[N[(N[(U / N[(J * 4.0), $MachinePrecision]), $MachinePrecision] * N[(U / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 70.2

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

5. Simplified 70.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. frac-times N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-/.f64 70.2

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 70.2

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

7. Applied egg-rr 70.2%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{1}{\frac{2 \cdot J}{U} \cdot \frac{2 \cdot J}{U}}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. pow2 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. +-commutative N/A

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

9. Applied egg-rr 70.2%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(J \cdot -2\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    2. div-inv N/A

       \[\leadsto \left(\left(J \cdot -2\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    3. metadata-eval N/A

       \[\leadsto \left(\left(J \cdot -2\right) \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    4. lift-*.f64 N/A

       \[\leadsto \left(\left(J \cdot -2\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    5. lift-cos.f64 N/A

       \[\leadsto \left(\left(J \cdot -2\right) \cdot \color{blue}{\cos \left(K \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    6. associate-*l* N/A

       \[\leadsto \color{blue}{\left(J \cdot \left(-2 \cdot \cos \left(K \cdot \frac{1}{2}\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    7. *-commutative N/A

       \[\leadsto \left(J \cdot \color{blue}{\left(\cos \left(K \cdot \frac{1}{2}\right) \cdot -2\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    8. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\cos \left(K \cdot \frac{1}{2}\right) \cdot -2\right) \cdot J\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    9. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(\cos \left(K \cdot \frac{1}{2}\right) \cdot -2\right) \cdot J\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    10. lower-*.f64 70.2

        \[\leadsto \left(\color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    11. lift-*.f64 N/A

        \[\leadsto \left(\left(\cos \color{blue}{\left(K \cdot \frac{1}{2}\right)} \cdot -2\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    12. *-commutative N/A

        \[\leadsto \left(\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot -2\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    13. lower-*.f64 70.2

        \[\leadsto \left(\left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot -2\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

11. Applied egg-rr 70.2%

    \[\leadsto \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot -2\right) \cdot J\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

12. Final simplification 70.2%

    \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot -2\right)\right) \]
13. Add Preprocessing

Alternative 13: 45.6% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (* (* J -2.0) (sqrt (fma (/ U (* J 4.0)) (/ U J) 1.0))))

C

double code(double J, double K, double U) {
	return (J * -2.0) * sqrt(fma((U / (J * 4.0)), (U / J), 1.0));
}

Julia

function code(J, K, U)
	return Float64(Float64(J * -2.0) * sqrt(fma(Float64(U / Float64(J * 4.0)), Float64(U / J), 1.0)))
end

Wolfram

code[J_, K_, U_] := N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(U / N[(J * 4.0), $MachinePrecision]), $MachinePrecision] * N[(U / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 70.2

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

5. Simplified 70.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. frac-times N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-/.f64 70.2

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 70.2

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

7. Applied egg-rr 70.2%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{1}{\frac{2 \cdot J}{U} \cdot \frac{2 \cdot J}{U}}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. pow2 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. +-commutative N/A

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

9. Applied egg-rr 70.2%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)}} \]

10. Taylor expanded in K around 0

    \[\leadsto \color{blue}{\left(-2 \cdot J\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

    2. lower-*.f64 47.1

       \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]

12. Simplified 47.1%

    \[\leadsto \color{blue}{\left(J \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 4}, \frac{U}{J}, 1\right)} \]
13. Add Preprocessing

Alternative 14: 30.3% accurate, 62.2× speedup

Math

\[\begin{array}{l} \\ J \cdot -2 \end{array} \]

FPCore

(FPCore (J K U) :precision binary64 (* J -2.0))

C

double code(double J, double K, double U) {
	return J * -2.0;
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = j * (-2.0d0)
end function

Java

public static double code(double J, double K, double U) {
	return J * -2.0;
}

Python

def code(J, K, U):
	return J * -2.0

Julia

function code(J, K, U)
	return Float64(J * -2.0)
end

MATLAB

function tmp = code(J, K, U)
	tmp = J * -2.0;
end

Wolfram

code[J_, K_, U_] := N[(J * -2.0), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in J around inf

   \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 55.8

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

5. Simplified 55.8%

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

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{-2 \cdot J} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{J \cdot -2} \]

   2. lower-*.f64 29.9

      \[\leadsto \color{blue}{J \cdot -2} \]

8. Simplified 29.9%

   \[\leadsto \color{blue}{J \cdot -2} \]
9. Add Preprocessing

Alternative 15: 9.5% accurate, 124.3× speedup

Math

\[\begin{array}{l} \\ -J \end{array} \]

FPCore

(FPCore (J K U) :precision binary64 (- J))

C

double code(double J, double K, double U) {
	return -J;
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = -j
end function

Java

public static double code(double J, double K, double U) {
	return -J;
}

Python

def code(J, K, U):
	return -J

Julia

function code(J, K, U)
	return Float64(-J)
end

MATLAB

function tmp = code(J, K, U)
	tmp = -J;
end

Wolfram

code[J_, K_, U_] := (-J)

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in J around inf

   \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 55.8

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

5. Simplified 55.8%

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

6. Taylor expanded in K around 0

   \[\leadsto \color{blue}{-2 \cdot J} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{J \cdot -2} \]

   2. lower-*.f64 29.9

      \[\leadsto \color{blue}{J \cdot -2} \]

8. Simplified 29.9%

   \[\leadsto \color{blue}{J \cdot -2} \]

9. Taylor expanded in J around -inf

   \[\leadsto \color{blue}{-1 \cdot J} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(J\right)} \]

    2. lower-neg.f64 9.3

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

11. Simplified 9.3%

    \[\leadsto \color{blue}{-J} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))))