Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.8% → 99.7%

Time: 7.9s

Alternatives: 9

Speedup: 0.4×

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: 72.8% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1}\\ t_2 := t\_1 \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m}, J\_m \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0)))
        (t_2 (* t_1 (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_2 -1e+307)
      (fma (/ J_m U_m) (* J_m -2.0) (- U_m))
      (if (<= t_2 2e+307) (* (* (* (cos (* -0.5 K)) J_m) -2.0) t_1) 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 t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0));
	double t_2 = t_1 * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_2 <= -1e+307) {
		tmp = fma((J_m / U_m), (J_m * -2.0), -U_m);
	} else if (t_2 <= 2e+307) {
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * t_1;
	} 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)
	t_0 = cos(Float64(K / 2.0))
	t_1 = sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0))
	t_2 = Float64(t_1 * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_2 <= -1e+307)
		tmp = fma(Float64(J_m / U_m), Float64(J_m * -2.0), Float64(-U_m));
	elseif (t_2 <= 2e+307)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J_m) * -2.0) * t_1);
	else
		tmp = 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -1e+307], N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, 2e+307], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * t$95$1), $MachinePrecision], U$95$m]]), $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))))) < -9.99999999999999986e306

   1. Initial program 8.8%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 50.3%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 50.3%

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

   if -9.99999999999999986e306 < (*.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))))) < 1.99999999999999997e307

   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-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.99999999999999997e307 < (*.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 5.8%

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

   3. Taylor expanded in 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 51.6

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

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{-U} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.9%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.0%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2} + 1} \cdot \left(t\_1 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m}, J\_m \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+174}:\\ \;\;\;\;t\_0 \cdot \left(J\_m \cdot -2\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J\_m}, \frac{U\_m \cdot U\_m}{J\_m}, 1\right)} \cdot \left(\left(t\_0 \cdot J\_m\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0) 1.0))
          (* t_1 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_2 -2e+301)
      (fma (/ J_m U_m) (* J_m -2.0) (- U_m))
      (if (<= t_2 -1e+174)
        (* t_0 (* J_m -2.0))
        (if (<= t_2 2e+307)
          (*
           (sqrt (fma (/ 0.25 J_m) (/ (* U_m U_m) J_m) 1.0))
           (* (* t_0 J_m) -2.0))
          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 t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = sqrt((pow((U_m / ((2.0 * J_m) * t_1)), 2.0) + 1.0)) * (t_1 * (J_m * -2.0));
	double tmp;
	if (t_2 <= -2e+301) {
		tmp = fma((J_m / U_m), (J_m * -2.0), -U_m);
	} else if (t_2 <= -1e+174) {
		tmp = t_0 * (J_m * -2.0);
	} else if (t_2 <= 2e+307) {
		tmp = sqrt(fma((0.25 / J_m), ((U_m * U_m) / J_m), 1.0)) * ((t_0 * J_m) * -2.0);
	} 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)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0) + 1.0)) * Float64(t_1 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_2 <= -2e+301)
		tmp = fma(Float64(J_m / U_m), Float64(J_m * -2.0), Float64(-U_m));
	elseif (t_2 <= -1e+174)
		tmp = Float64(t_0 * Float64(J_m * -2.0));
	elseif (t_2 <= 2e+307)
		tmp = Float64(sqrt(fma(Float64(0.25 / J_m), Float64(Float64(U_m * U_m) / J_m), 1.0)) * Float64(Float64(t_0 * J_m) * -2.0));
	else
		tmp = 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_] := 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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+301], N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, -1e+174], N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+307], N[(N[Sqrt[N[(N[(0.25 / J$95$m), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], U$95$m]]]), $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))))) < -2.00000000000000011e301

   1. Initial program 20.0%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 44.5

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

   5. Applied rewrites 44.5%

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

   7. Applied rewrites 46.6%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 46.6%

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

   if -2.00000000000000011e301 < (*.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))))) < -1.00000000000000007e174

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 88.1

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

   7. Applied rewrites 88.1%

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

   if -1.00000000000000007e174 < (*.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))))) < 1.99999999999999997e307

   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-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 78.2

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

   7. Applied rewrites 78.2%

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

   if 1.99999999999999997e307 < (*.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 5.8%

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

   3. Taylor expanded in 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 51.6

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

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{-U} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.9%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{+174}:\\ \;\;\;\;\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)} \cdot \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \cos \left(-0.5 \cdot K\right) \cdot J\_m\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m}, J\_m \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, {\left(\frac{t\_1}{U\_m}\right)}^{-2}, 1\right)} \cdot \left(t\_1 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (cos (* -0.5 K)) J_m))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_2 -2e+301)
      (fma (/ J_m U_m) (* J_m -2.0) (- U_m))
      (if (<= t_2 2e+307)
        (* (sqrt (fma 0.25 (pow (/ t_1 U_m) -2.0) 1.0)) (* t_1 -2.0))
        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 t_0 = cos((K / 2.0));
	double t_1 = cos((-0.5 * K)) * J_m;
	double t_2 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_2 <= -2e+301) {
		tmp = fma((J_m / U_m), (J_m * -2.0), -U_m);
	} else if (t_2 <= 2e+307) {
		tmp = sqrt(fma(0.25, pow((t_1 / U_m), -2.0), 1.0)) * (t_1 * -2.0);
	} 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)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(cos(Float64(-0.5 * K)) * J_m)
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_2 <= -2e+301)
		tmp = fma(Float64(J_m / U_m), Float64(J_m * -2.0), Float64(-U_m));
	elseif (t_2 <= 2e+307)
		tmp = Float64(sqrt(fma(0.25, (Float64(t_1 / U_m) ^ -2.0), 1.0)) * Float64(t_1 * -2.0));
	else
		tmp = 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+301], N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, 2e+307], N[(N[Sqrt[N[(0.25 * N[Power[N[(t$95$1 / U$95$m), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision], U$95$m]]), $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))))) < -2.00000000000000011e301

   1. Initial program 20.0%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 44.5

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

   5. Applied rewrites 44.5%

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

   7. Applied rewrites 46.6%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 46.6%

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

   if -2.00000000000000011e301 < (*.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))))) < 1.99999999999999997e307

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

   8. Applied rewrites 99.7%

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

   if 1.99999999999999997e307 < (*.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 5.8%

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

   3. Taylor expanded in 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 51.6

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

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{-U} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.9%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.0%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, {\left(\frac{\cos \left(-0.5 \cdot K\right) \cdot J}{U}\right)}^{-2}, 1\right)} \cdot \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m}, J\_m \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\cos \left(-0.5 \cdot K\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -2e+301)
      (fma (/ J_m U_m) (* J_m -2.0) (- U_m))
      (if (<= t_1 2e+307) (* (cos (* -0.5 K)) (* J_m -2.0)) 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 t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -2e+301) {
		tmp = fma((J_m / U_m), (J_m * -2.0), -U_m);
	} else if (t_1 <= 2e+307) {
		tmp = cos((-0.5 * K)) * (J_m * -2.0);
	} 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)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -2e+301)
		tmp = fma(Float64(J_m / U_m), Float64(J_m * -2.0), Float64(-U_m));
	elseif (t_1 <= 2e+307)
		tmp = Float64(cos(Float64(-0.5 * K)) * Float64(J_m * -2.0));
	else
		tmp = 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+301], N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, 2e+307], N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], U$95$m]]), $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))))) < -2.00000000000000011e301

   1. Initial program 20.0%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 44.5

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

   5. Applied rewrites 44.5%

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

   7. Applied rewrites 46.6%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 46.6%

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

   if -2.00000000000000011e301 < (*.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))))) < 1.99999999999999997e307

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

   if 1.99999999999999997e307 < (*.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 5.8%

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

   3. Taylor expanded in 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 51.6

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

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{-U} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.9%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.0%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m}, J\_m \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J\_m}, \frac{U\_m \cdot U\_m}{J\_m}, 1\right)} \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -5e+278)
      (fma (/ J_m U_m) (* J_m -2.0) (- U_m))
      (if (<= t_1 -1e-270)
        (* (sqrt (fma (/ 0.25 J_m) (/ (* U_m U_m) J_m) 1.0)) (* J_m -2.0))
        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 t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -5e+278) {
		tmp = fma((J_m / U_m), (J_m * -2.0), -U_m);
	} else if (t_1 <= -1e-270) {
		tmp = sqrt(fma((0.25 / J_m), ((U_m * U_m) / J_m), 1.0)) * (J_m * -2.0);
	} 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)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -5e+278)
		tmp = fma(Float64(J_m / U_m), Float64(J_m * -2.0), Float64(-U_m));
	elseif (t_1 <= -1e-270)
		tmp = Float64(sqrt(fma(Float64(0.25 / J_m), Float64(Float64(U_m * U_m) / J_m), 1.0)) * Float64(J_m * -2.0));
	else
		tmp = 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+278], N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, -1e-270], N[(N[Sqrt[N[(N[(0.25 / J$95$m), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], U$95$m]]), $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))))) < -5.00000000000000029e278

   1. Initial program 28.6%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 40.3

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

   5. Applied rewrites 40.3%

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 42.2%

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

   if -5.00000000000000029e278 < (*.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))))) < -1e-270

   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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 53.3

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

   5. Applied rewrites 53.3%

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

   if -1e-270 < (*.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 70.1%

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

   3. 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 29.0

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

   5. Applied rewrites 29.0%

      \[\leadsto \color{blue}{-U} \]
   6. Step-by-step derivation

   7. Applied rewrites 15.4%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.6%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m}, J\_m \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-270}:\\ \;\;\;\;1 \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -2e+301)
      (fma (/ J_m U_m) (* J_m -2.0) (- U_m))
      (if (<= t_1 -1e-270) (* 1.0 (* J_m -2.0)) 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 t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -2e+301) {
		tmp = fma((J_m / U_m), (J_m * -2.0), -U_m);
	} else if (t_1 <= -1e-270) {
		tmp = 1.0 * (J_m * -2.0);
	} 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)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -2e+301)
		tmp = fma(Float64(J_m / U_m), Float64(J_m * -2.0), Float64(-U_m));
	elseif (t_1 <= -1e-270)
		tmp = Float64(1.0 * Float64(J_m * -2.0));
	else
		tmp = 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+301], N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, -1e-270], N[(1.0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], U$95$m]]), $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))))) < -2.00000000000000011e301

   1. Initial program 20.0%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 44.5

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

   5. Applied rewrites 44.5%

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

   7. Applied rewrites 46.6%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 46.6%

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

   if -2.00000000000000011e301 < (*.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))))) < -1e-270

   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 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 76.8%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 44.7

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

   8. Applied rewrites 44.7%

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

   if -1e-270 < (*.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 70.1%

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

   3. 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 29.0

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

   5. Applied rewrites 29.0%

      \[\leadsto \color{blue}{-U} \]
   6. Step-by-step derivation

   7. Applied rewrites 15.4%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.6%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-270}:\\ \;\;\;\;1 \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+301}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-270}:\\ \;\;\;\;1 \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -2e+301)
      (- U_m)
      (if (<= t_1 -1e-270) (* 1.0 (* J_m -2.0)) 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 t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -2e+301) {
		tmp = -U_m;
	} else if (t_1 <= -1e-270) {
		tmp = 1.0 * (J_m * -2.0);
	} else {
		tmp = U_m;
	}
	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) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = sqrt((((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0) + 1.0d0)) * (t_0 * (j_m * (-2.0d0)))
    if (t_1 <= (-2d+301)) then
        tmp = -u_m
    else if (t_1 <= (-1d-270)) then
        tmp = 1.0d0 * (j_m * (-2.0d0))
    else
        tmp = u_m
    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((K / 2.0));
	double t_1 = Math.sqrt((Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -2e+301) {
		tmp = -U_m;
	} else if (t_1 <= -1e-270) {
		tmp = 1.0 * (J_m * -2.0);
	} else {
		tmp = U_m;
	}
	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((K / 2.0))
	t_1 = math.sqrt((math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0))
	tmp = 0
	if t_1 <= -2e+301:
		tmp = -U_m
	elif t_1 <= -1e-270:
		tmp = 1.0 * (J_m * -2.0)
	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)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -2e+301)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-270)
		tmp = Float64(1.0 * Float64(J_m * -2.0));
	else
		tmp = U_m;
	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((K / 2.0));
	t_1 = sqrt((((U_m / ((2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	tmp = 0.0;
	if (t_1 <= -2e+301)
		tmp = -U_m;
	elseif (t_1 <= -1e-270)
		tmp = 1.0 * (J_m * -2.0);
	else
		tmp = U_m;
	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[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+301], (-U$95$m), If[LessEqual[t$95$1, -1e-270], N[(1.0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], U$95$m]]), $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))))) < -2.00000000000000011e301

   1. Initial program 20.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 46.7

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

   5. Applied rewrites 46.7%

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

   if -2.00000000000000011e301 < (*.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))))) < -1e-270

   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 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 76.8%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 44.7

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

   8. Applied rewrites 44.7%

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

   if -1e-270 < (*.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 70.1%

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

   3. 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 29.0

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

   5. Applied rewrites 29.0%

      \[\leadsto \color{blue}{-U} \]
   6. Step-by-step derivation

   7. Applied rewrites 15.4%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.6%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+301}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-270}:\\ \;\;\;\;1 \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right) \leq -1 \cdot 10^{-270}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (<=
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))
         -1e-270)
      (- U_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 t_0 = cos((K / 2.0));
	double tmp;
	if ((sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0))) <= -1e-270) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	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) :: tmp
    t_0 = cos((k / 2.0d0))
    if ((sqrt((((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0) + 1.0d0)) * (t_0 * (j_m * (-2.0d0)))) <= (-1d-270)) then
        tmp = -u_m
    else
        tmp = u_m
    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((K / 2.0));
	double tmp;
	if ((Math.sqrt((Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0))) <= -1e-270) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	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((K / 2.0))
	tmp = 0
	if (math.sqrt((math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0))) <= -1e-270:
		tmp = -U_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)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0))) <= -1e-270)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	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((K / 2.0));
	tmp = 0.0;
	if ((sqrt((((U_m / ((2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * (t_0 * (J_m * -2.0))) <= -1e-270)
		tmp = -U_m;
	else
		tmp = U_m;
	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[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-270], (-U$95$m), U$95$m]), $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))))) < -1e-270

   1. Initial program 76.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 25.3

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

   5. Applied rewrites 25.3%

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

   if -1e-270 < (*.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 70.1%

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

   3. 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 29.0

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

   5. Applied rewrites 29.0%

      \[\leadsto \color{blue}{-U} \]
   6. Step-by-step derivation

   7. Applied rewrites 15.4%

      \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.6%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-270}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 14.1% accurate, 373.0× speedup

Math

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

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 * 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 73.4%

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

3. Taylor expanded in J around 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 27.0

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

5. Applied rewrites 27.0%

   \[\leadsto \color{blue}{-U} \]
6. Step-by-step derivation

7. Applied rewrites 13.9%

   \[\leadsto \frac{0 - U \cdot U}{\color{blue}{0 + U}} \]
8. Step-by-step derivation

9. Applied rewrites 24.7%

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

Reproduce

herbie shell --seed 2024312 
(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)))))