Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 98.8%

Time: 8.9s

Alternatives: 11

Speedup: 0.7×

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: 73.3% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -3 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+287}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{t\_0} \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 -3e+306)
      (- U_m)
      (if (<= t_2 1e+287) t_2 (* (/ U_m t_0) t_0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -3e+306) {
		tmp = -U_m;
	} else if (t_2 <= 1e+287) {
		tmp = t_2;
	} else {
		tmp = (U_m / t_0) * t_0;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(((-0.5d0) * k))
    t_1 = cos((k / 2.0d0))
    t_2 = (((-2.0d0) * j_m) * t_1) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_1)) ** 2.0d0)))
    if (t_2 <= (-3d+306)) then
        tmp = -u_m
    else if (t_2 <= 1d+287) then
        tmp = t_2
    else
        tmp = (u_m / t_0) * t_0
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -3e+306) {
		tmp = -U_m;
	} else if (t_2 <= 1e+287) {
		tmp = t_2;
	} else {
		tmp = (U_m / t_0) * t_0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -3e+306:
		tmp = -U_m
	elif t_2 <= 1e+287:
		tmp = t_2
	else:
		tmp = (U_m / t_0) * t_0
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -3e+306)
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+287)
		tmp = t_2;
	else
		tmp = Float64(Float64(U_m / t_0) * t_0);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((-0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -3e+306)
		tmp = -U_m;
	elseif (t_2 <= 1e+287)
		tmp = t_2;
	else
		tmp = (U_m / t_0) * t_0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -3e+306], (-U$95$m), If[LessEqual[t$95$2, 1e+287], t$95$2, N[(N[(U$95$m / t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.00000000000000021e306

   1. Initial program 8.0%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -3.00000000000000021e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.0000000000000001e287

   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

   if 1.0000000000000001e287 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 12.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 12.1

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 12.1

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

   4. Applied rewrites 12.1%

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

   6. Applied rewrites 12.1%

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

   7. Taylor expanded in U around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 39.6

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

   9. Applied rewrites 39.6%

      \[\leadsto \color{blue}{\frac{U}{\cos \left(-0.5 \cdot K\right)}} \cdot \cos \left(-0.5 \cdot K\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ t_3 := t\_2 \cdot \left(-2 \cdot J\_m\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -3 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+171}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-101}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 1\right)} \cdot \left(J\_m \cdot -2\right)\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (cos (* -0.5 K)))
        (t_3 (* t_2 (* -2.0 J_m))))
   (*
    J_s
    (if (<= t_1 -3e+306)
      (- U_m)
      (if (<= t_1 -5e+171)
        t_3
        (if (<= t_1 -5e-101)
          (*
           (* (sqrt (fma 0.25 (/ (* U_m U_m) (* J_m J_m)) 1.0)) (* J_m -2.0))
           t_2)
          (if (<= t_1 -4e-286)
            (* (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)) (* -2.0 J_m))
            t_3)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = cos((-0.5 * K));
	double t_3 = t_2 * (-2.0 * J_m);
	double tmp;
	if (t_1 <= -3e+306) {
		tmp = -U_m;
	} else if (t_1 <= -5e+171) {
		tmp = t_3;
	} else if (t_1 <= -5e-101) {
		tmp = (sqrt(fma(0.25, ((U_m * U_m) / (J_m * J_m)), 1.0)) * (J_m * -2.0)) * t_2;
	} else if (t_1 <= -4e-286) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * (-2.0 * J_m);
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(-0.5 * K))
	t_3 = Float64(t_2 * Float64(-2.0 * J_m))
	tmp = 0.0
	if (t_1 <= -3e+306)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e+171)
		tmp = t_3;
	elseif (t_1 <= -5e-101)
		tmp = Float64(Float64(sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 1.0)) * Float64(J_m * -2.0)) * t_2);
	elseif (t_1 <= -4e-286)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(-2.0 * J_m));
	else
		tmp = t_3;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -3e+306], (-U$95$m), If[LessEqual[t$95$1, -5e+171], t$95$3, If[LessEqual[t$95$1, -5e-101], N[(N[(N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, -4e-286], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], t$95$3]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.00000000000000021e306

   1. Initial program 8.0%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -3.00000000000000021e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000004e171 or -4.0000000000000002e-286 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 75.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 74.9

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 74.9

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

   4. Applied rewrites 74.9%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 55.3

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

   9. Applied rewrites 55.3%

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

   if -5.0000000000000004e171 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000001e-101

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in K around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 93.2

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

   9. Applied rewrites 93.2%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{U}{J} \cdot \frac{U}{J}}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(-0.5 \cdot K\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 91.2%

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

   if -5.0000000000000001e-101 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000002e-286

   1. Initial program 99.4%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(-2 \cdot J\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 78.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -3 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+193} \lor \neg \left(t\_1 \leq -4 \cdot 10^{-286}\right):\\ \;\;\;\;\cos \left(-0.5 \cdot K\right) \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -3e+306)
      (- U_m)
      (if (or (<= t_1 -5e+193) (not (<= t_1 -4e-286)))
        (* (cos (* -0.5 K)) (* -2.0 J_m))
        (* (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)) (* -2.0 J_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -3e+306) {
		tmp = -U_m;
	} else if ((t_1 <= -5e+193) || !(t_1 <= -4e-286)) {
		tmp = cos((-0.5 * K)) * (-2.0 * J_m);
	} else {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * (-2.0 * J_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -3e+306)
		tmp = Float64(-U_m);
	elseif ((t_1 <= -5e+193) || !(t_1 <= -4e-286))
		tmp = Float64(cos(Float64(-0.5 * K)) * Float64(-2.0 * J_m));
	else
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(-2.0 * J_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -3e+306], (-U$95$m), If[Or[LessEqual[t$95$1, -5e+193], N[Not[LessEqual[t$95$1, -4e-286]], $MachinePrecision]], N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.00000000000000021e306

   1. Initial program 8.0%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -3.00000000000000021e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999972e193 or -4.0000000000000002e-286 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 74.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 74.4

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 74.4

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

   4. Applied rewrites 74.4%

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

   6. Applied rewrites 74.4%

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

   7. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 55.7

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

   9. Applied rewrites 55.7%

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

   if -4.99999999999999972e193 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000002e-286

   1. Initial program 99.7%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 58.9

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

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(-2 \cdot J\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -3 \cdot 10^{+306}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -5 \cdot 10^{+193} \lor \neg \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -4 \cdot 10^{-286}\right):\\ \;\;\;\;\cos \left(-0.5 \cdot K\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(-2 \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+287}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(0.25, {\left(\frac{J\_m}{U\_m} \cdot t\_0\right)}^{-2}, 1\right)} \cdot \left(J\_m \cdot -2\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{t\_0} \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 1e+287)
        (*
         (* (sqrt (fma 0.25 (pow (* (/ J_m U_m) t_0) -2.0) 1.0)) (* J_m -2.0))
         t_0)
        (* (/ U_m t_0) t_0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+287) {
		tmp = (sqrt(fma(0.25, pow(((J_m / U_m) * t_0), -2.0), 1.0)) * (J_m * -2.0)) * t_0;
	} else {
		tmp = (U_m / t_0) * t_0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+287)
		tmp = Float64(Float64(sqrt(fma(0.25, (Float64(Float64(J_m / U_m) * t_0) ^ -2.0), 1.0)) * Float64(J_m * -2.0)) * t_0);
	else
		tmp = Float64(Float64(U_m / t_0) * t_0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+287], N[(N[(N[Sqrt[N[(0.25 * N[Power[N[(N[(J$95$m / U$95$m), $MachinePrecision] * t$95$0), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(U$95$m / t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 5.3%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 50.4

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

   5. Applied rewrites 50.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.0000000000000001e287

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.6

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

   if 1.0000000000000001e287 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 12.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 12.1

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 12.1

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

   4. Applied rewrites 12.1%

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

   6. Applied rewrites 12.1%

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

   7. Taylor expanded in U around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 39.6

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

   9. Applied rewrites 39.6%

      \[\leadsto \color{blue}{\frac{U}{\cos \left(-0.5 \cdot K\right)}} \cdot \cos \left(-0.5 \cdot K\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -3 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+287}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(0.25, \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 1\right)} \cdot \left(J\_m \cdot -2\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{t\_0} \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 -3e+306)
      (- U_m)
      (if (<= t_2 1e+287)
        (*
         (* (sqrt (fma 0.25 (* (/ U_m J_m) (/ U_m J_m)) 1.0)) (* J_m -2.0))
         t_0)
        (* (/ U_m t_0) t_0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -3e+306) {
		tmp = -U_m;
	} else if (t_2 <= 1e+287) {
		tmp = (sqrt(fma(0.25, ((U_m / J_m) * (U_m / J_m)), 1.0)) * (J_m * -2.0)) * t_0;
	} else {
		tmp = (U_m / t_0) * t_0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -3e+306)
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+287)
		tmp = Float64(Float64(sqrt(fma(0.25, Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 1.0)) * Float64(J_m * -2.0)) * t_0);
	else
		tmp = Float64(Float64(U_m / t_0) * t_0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -3e+306], (-U$95$m), If[LessEqual[t$95$2, 1e+287], N[(N[(N[Sqrt[N[(0.25 * N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(U$95$m / t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.00000000000000021e306

   1. Initial program 8.0%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -3.00000000000000021e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.0000000000000001e287

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.6

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in K around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 89.3

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

   9. Applied rewrites 89.3%

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

   if 1.0000000000000001e287 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 12.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 12.1

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 12.1

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

   4. Applied rewrites 12.1%

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

   6. Applied rewrites 12.1%

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

   7. Taylor expanded in U around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 39.6

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

   9. Applied rewrites 39.6%

      \[\leadsto \color{blue}{\frac{U}{\cos \left(-0.5 \cdot K\right)}} \cdot \cos \left(-0.5 \cdot K\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -3 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+287}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(0.25, \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 1\right)} \cdot \left(J\_m \cdot -2\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot K\right) \cdot U\_m}{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 -3e+306)
      (- U_m)
      (if (<= t_2 1e+287)
        (*
         (* (sqrt (fma 0.25 (* (/ U_m J_m) (/ U_m J_m)) 1.0)) (* J_m -2.0))
         t_0)
        (/ (* (cos (* 0.5 K)) U_m) t_0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -3e+306) {
		tmp = -U_m;
	} else if (t_2 <= 1e+287) {
		tmp = (sqrt(fma(0.25, ((U_m / J_m) * (U_m / J_m)), 1.0)) * (J_m * -2.0)) * t_0;
	} else {
		tmp = (cos((0.5 * K)) * U_m) / t_0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -3e+306)
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+287)
		tmp = Float64(Float64(sqrt(fma(0.25, Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 1.0)) * Float64(J_m * -2.0)) * t_0);
	else
		tmp = Float64(Float64(cos(Float64(0.5 * K)) * U_m) / t_0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -3e+306], (-U$95$m), If[LessEqual[t$95$2, 1e+287], N[(N[(N[Sqrt[N[(0.25 * N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * U$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.00000000000000021e306

   1. Initial program 8.0%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -3.00000000000000021e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.0000000000000001e287

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.6

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in K around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 89.3

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

   9. Applied rewrites 89.3%

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

   if 1.0000000000000001e287 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 12.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 12.1

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 12.1

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

   4. Applied rewrites 12.1%

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

   5. Taylor expanded in U around -inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.5

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

   7. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(-0.5 \cdot K\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}} \leq -3 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(0.25, \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 1\right)} \cdot \left(J\_m \cdot -2\right)\right) \cdot \cos \left(-0.5 \cdot K\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
         -3e+306)
      (- U_m)
      (*
       (* (sqrt (fma 0.25 (* (/ U_m J_m) (/ U_m J_m)) 1.0)) (* J_m -2.0))
       (cos (* -0.5 K)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -3e+306) {
		tmp = -U_m;
	} else {
		tmp = (sqrt(fma(0.25, ((U_m / J_m) * (U_m / J_m)), 1.0)) * (J_m * -2.0)) * cos((-0.5 * K));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0)))) <= -3e+306)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(sqrt(fma(0.25, Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 1.0)) * Float64(J_m * -2.0)) * cos(Float64(-0.5 * K)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -3e+306], (-U$95$m), N[(N[(N[Sqrt[N[(0.25 * N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.00000000000000021e306

   1. Initial program 8.0%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -3.00000000000000021e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 83.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 83.4

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

      14. lift-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac2 N/A

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

      18. cos-neg N/A

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

      19. lower-cos.f64 N/A

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

      20. div-inv N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 83.4

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

   4. Applied rewrites 83.4%

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

   6. Applied rewrites 83.4%

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

   7. Taylor expanded in K around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 75.0

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

   9. Applied rewrites 75.0%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{U}{J} \cdot \frac{U}{J}}, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(-0.5 \cdot K\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
         (- INFINITY))
      (- U_m)
      (* (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)) (* -2.0 J_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -((double) INFINITY)) {
		tmp = -U_m;
	} else {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * (-2.0 * J_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(-U_m);
	else
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(-2.0 * J_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], (-U$95$m), N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 5.3%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 50.4

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

   5. Applied rewrites 50.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 83.6%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 49.3

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

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(-2 \cdot J\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 49.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m}, -0.25, -2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 2.7e+19) (fma (/ (* U_m U_m) J_m) -0.25 (* -2.0 J_m)) (- U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 2.7e+19) {
		tmp = fma(((U_m * U_m) / J_m), -0.25, (-2.0 * J_m));
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 2.7e+19)
		tmp = fma(Float64(Float64(U_m * U_m) / J_m), -0.25, Float64(-2.0 * J_m));
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 2.7e+19], N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * -0.25 + N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], (-U$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 2.7e19

   1. Initial program 79.5%

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

   3. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in K around 0

      \[\leadsto -2 \cdot J + \color{blue}{\frac{-1}{4} \cdot \frac{{U}^{2}}{J}} \]
   7. Step-by-step derivation

   8. Applied rewrites 32.0%

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

   if 2.7e19 < U

   1. Initial program 52.6%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 44.7

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

   5. Applied rewrites 44.7%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 47.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(0.25 \cdot \left(K \cdot K\right), J\_m, -2 \cdot J\_m\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 2.7e+19)
    (* (fma (* 0.25 (* K K)) J_m (* -2.0 J_m)) 1.0)
    (- U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 2.7e+19) {
		tmp = fma((0.25 * (K * K)), J_m, (-2.0 * J_m)) * 1.0;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 2.7e+19)
		tmp = Float64(fma(Float64(0.25 * Float64(K * K)), J_m, Float64(-2.0 * J_m)) * 1.0);
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 2.7e+19], N[(N[(N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision] * J$95$m + N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], (-U$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 2.7e19

   1. Initial program 79.5%

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

   3. Taylor expanded in J around inf

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 29.1

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

   8. Applied rewrites 29.1%

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

   if 2.7e19 < U

   1. Initial program 52.6%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 44.7

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

   5. Applied rewrites 44.7%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 39.2% accurate, 124.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(-U\_m\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (- U_m)))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * -U_m;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * -u_m
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * -U_m;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * -U_m

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(-U_m))
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * -U_m;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * (-U$95$m)), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

3. Taylor expanded in J around 0

   \[\leadsto \color{blue}{-1 \cdot U} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

   2. lower-neg.f64 28.6

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

5. Applied rewrites 28.6%

   \[\leadsto \color{blue}{-U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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)))))