Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.0% → 99.5%

Time: 12.8s

Alternatives: 14

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: 73.0% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;t\_1 \cdot \left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))
        (t_2 (* (* (* -2.0 J) t_0) t_1)))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 1e+304) (* t_1 (* -2.0 (* J (cos (* K 0.5))))) U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = ((-2.0 * J) * t_0) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 * (-2.0 * (J * cos((K * 0.5))));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = ((-2.0 * J) * t_0) * t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 * (-2.0 * (J * Math.cos((K * 0.5))));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))
	t_2 = ((-2.0 * J) * t_0) * t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 1e+304:
		tmp = t_1 * (-2.0 * (J * math.cos((K * 0.5))))
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0)))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_0) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+304)
		tmp = Float64(t_1 * Float64(-2.0 * Float64(J * cos(Float64(K * 0.5)))));
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)));
	t_2 = ((-2.0 * J) * t_0) * t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 1e+304)
		tmp = t_1 * (-2.0 * (J * cos((K * 0.5))));
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+304], N[(t$95$1 * N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999994e303

   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

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 9.9999999999999994e303 < (*.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 5.4%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 71.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.2

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

   10. Applied rewrites 71.2%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+304}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \cdot \left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot J\right)}, 1\right)}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_3 := \left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{J \cdot 2}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-185}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-U\_m\right) \cdot \left(-1 + -2 \cdot \left(0.5 \cdot \cos K\right)\right)\right) \cdot \frac{J}{U\_m}, \frac{J}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0
         (*
          (* -2.0 (* J (cos (* K 0.5))))
          (sqrt
           (fma 0.25 (/ (* U_m U_m) (* (fma 0.5 (cos K) 0.5) (* J J))) 1.0))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0)))))
        (t_3 (* (* -2.0 J) (sqrt (+ 1.0 (pow (/ U_m (* J 2.0)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -5e+202)
       t_3
       (if (<= t_2 -5e-39)
         t_0
         (if (<= t_2 2e-185)
           t_3
           (if (<= t_2 2e+299)
             t_0
             (fma
              (* (* (- U_m) (+ -1.0 (* -2.0 (* 0.5 (cos K))))) (/ J U_m))
              (/ J U_m)
              U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = (-2.0 * (J * cos((K * 0.5)))) * sqrt(fma(0.25, ((U_m * U_m) / (fma(0.5, cos(K), 0.5) * (J * J))), 1.0));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double t_3 = (-2.0 * J) * sqrt((1.0 + pow((U_m / (J * 2.0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -5e+202) {
		tmp = t_3;
	} else if (t_2 <= -5e-39) {
		tmp = t_0;
	} else if (t_2 <= 2e-185) {
		tmp = t_3;
	} else if (t_2 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = fma(((-U_m * (-1.0 + (-2.0 * (0.5 * cos(K))))) * (J / U_m)), (J / U_m), U_m);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(Float64(-2.0 * Float64(J * cos(Float64(K * 0.5)))) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(fma(0.5, cos(K), 0.5) * Float64(J * J))), 1.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	t_3 = Float64(Float64(-2.0 * J) * sqrt(Float64(1.0 + (Float64(U_m / Float64(J * 2.0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -5e+202)
		tmp = t_3;
	elseif (t_2 <= -5e-39)
		tmp = t_0;
	elseif (t_2 <= 2e-185)
		tmp = t_3;
	elseif (t_2 <= 2e+299)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(Float64(-U_m) * Float64(-1.0 + Float64(-2.0 * Float64(0.5 * cos(K))))) * Float64(J / U_m)), Float64(J / U_m), U_m);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision] * N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -5e+202], t$95$3, If[LessEqual[t$95$2, -5e-39], t$95$0, If[LessEqual[t$95$2, 2e-185], t$95$3, If[LessEqual[t$95$2, 2e+299], t$95$0, N[(N[(N[((-U$95$m) * N[(-1.0 + N[(-2.0 * N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] * N[(J / U$95$m), $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))))) < -inf.0

   1. Initial program 6.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999999e202 or -4.9999999999999998e-39 < (*.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))))) < 2e-185

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -4.9999999999999999e202 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999998e-39 or 2e-185 < (*.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))))) < 2.0000000000000001e299

   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

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 95.1%

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

   7. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.7

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

   9. Applied rewrites 83.7%

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

   if 2.0000000000000001e299 < (*.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 11.1%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 67.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-39}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot J\right)}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{-185}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot J\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-U\right) \cdot \left(-1 + -2 \cdot \left(0.5 \cdot \cos K\right)\right)\right) \cdot \frac{J}{U}, \frac{J}{U}, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := J \cdot \left(\sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot J\right)}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_3 := \left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{J \cdot 2}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-185}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-U\_m\right) \cdot \left(-1 + -2 \cdot \left(0.5 \cdot \cos K\right)\right)\right) \cdot \frac{J}{U\_m}, \frac{J}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0
         (*
          J
          (*
           (sqrt
            (fma 0.25 (/ (* U_m U_m) (* (fma 0.5 (cos K) 0.5) (* J J))) 1.0))
           (* -2.0 (cos (* K 0.5))))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0)))))
        (t_3 (* (* -2.0 J) (sqrt (+ 1.0 (pow (/ U_m (* J 2.0)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -5e+202)
       t_3
       (if (<= t_2 -5e-39)
         t_0
         (if (<= t_2 2e-185)
           t_3
           (if (<= t_2 2e+299)
             t_0
             (fma
              (* (* (- U_m) (+ -1.0 (* -2.0 (* 0.5 (cos K))))) (/ J U_m))
              (/ J U_m)
              U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = J * (sqrt(fma(0.25, ((U_m * U_m) / (fma(0.5, cos(K), 0.5) * (J * J))), 1.0)) * (-2.0 * cos((K * 0.5))));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double t_3 = (-2.0 * J) * sqrt((1.0 + pow((U_m / (J * 2.0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -5e+202) {
		tmp = t_3;
	} else if (t_2 <= -5e-39) {
		tmp = t_0;
	} else if (t_2 <= 2e-185) {
		tmp = t_3;
	} else if (t_2 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = fma(((-U_m * (-1.0 + (-2.0 * (0.5 * cos(K))))) * (J / U_m)), (J / U_m), U_m);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(J * Float64(sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(fma(0.5, cos(K), 0.5) * Float64(J * J))), 1.0)) * Float64(-2.0 * cos(Float64(K * 0.5)))))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	t_3 = Float64(Float64(-2.0 * J) * sqrt(Float64(1.0 + (Float64(U_m / Float64(J * 2.0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -5e+202)
		tmp = t_3;
	elseif (t_2 <= -5e-39)
		tmp = t_0;
	elseif (t_2 <= 2e-185)
		tmp = t_3;
	elseif (t_2 <= 2e+299)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(Float64(-U_m) * Float64(-1.0 + Float64(-2.0 * Float64(0.5 * cos(K))))) * Float64(J / U_m)), Float64(J / U_m), U_m);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(J * N[(N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision] * N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -5e+202], t$95$3, If[LessEqual[t$95$2, -5e-39], t$95$0, If[LessEqual[t$95$2, 2e-185], t$95$3, If[LessEqual[t$95$2, 2e+299], t$95$0, N[(N[(N[((-U$95$m) * N[(-1.0 + N[(-2.0 * N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] * N[(J / U$95$m), $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))))) < -inf.0

   1. Initial program 6.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999999e202 or -4.9999999999999998e-39 < (*.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))))) < 2e-185

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -4.9999999999999999e202 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999998e-39 or 2e-185 < (*.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))))) < 2.0000000000000001e299

   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

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 95.1%

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

   7. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 83.7

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

   9. Applied rewrites 83.7%

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

       1. lift-*.f64 N/A

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

       2. lift-cos.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-cos.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-fma.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

   11. Applied rewrites 83.6%

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

   if 2.0000000000000001e299 < (*.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 11.1%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 67.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-39}:\\ \;\;\;\;J \cdot \left(\sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot J\right)}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{-185}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;J \cdot \left(\sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot J\right)}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-U\right) \cdot \left(-1 + -2 \cdot \left(0.5 \cdot \cos K\right)\right)\right) \cdot \frac{J}{U}, \frac{J}{U}, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(\left(-U\_m\right) \cdot \left(-1 + -2 \cdot \left(0.5 \cdot \cos K\right)\right)\right) \cdot \frac{J}{U\_m}, \frac{J}{U\_m}, U\_m\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_2 \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J \cdot J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0
         (fma
          (* (* (- U_m) (+ -1.0 (* -2.0 (* 0.5 (cos K))))) (/ J U_m))
          (/ J U_m)
          U_m))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0))))))
   (if (<= t_3 (- INFINITY))
     (- U_m)
     (if (<= t_3 -1e-306)
       (* (* -2.0 J) (sqrt (+ 1.0 (pow (/ U_m (* J 2.0)) 2.0))))
       (if (<= t_3 2e-94)
         t_0
         (if (<= t_3 2e+299)
           (* t_2 (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
           t_0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = fma(((-U_m * (-1.0 + (-2.0 * (0.5 * cos(K))))) * (J / U_m)), (J / U_m), U_m);
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_3 <= -1e-306) {
		tmp = (-2.0 * J) * sqrt((1.0 + pow((U_m / (J * 2.0)), 2.0)));
	} else if (t_3 <= 2e-94) {
		tmp = t_0;
	} else if (t_3 <= 2e+299) {
		tmp = t_2 * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = fma(Float64(Float64(Float64(-U_m) * Float64(-1.0 + Float64(-2.0 * Float64(0.5 * cos(K))))) * Float64(J / U_m)), Float64(J / U_m), U_m)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_3 <= -1e-306)
		tmp = Float64(Float64(-2.0 * J) * sqrt(Float64(1.0 + (Float64(U_m / Float64(J * 2.0)) ^ 2.0))));
	elseif (t_3 <= 2e-94)
		tmp = t_0;
	elseif (t_3 <= 2e+299)
		tmp = Float64(t_2 * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J * J)), 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(N[((-U$95$m) * N[(-1.0 + N[(-2.0 * N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], (-U$95$m), If[LessEqual[t$95$3, -1e-306], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-94], t$95$0, If[LessEqual[t$95$3, 2e+299], N[(t$95$2 * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]

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))))) < -inf.0

   1. Initial program 6.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.00000000000000003e-306

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 52.3

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

   8. Applied rewrites 52.3%

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

   if -1.00000000000000003e-306 < (*.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))))) < 1.9999999999999999e-94 or 2.0000000000000001e299 < (*.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 45.6%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 43.5

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

   5. Applied rewrites 43.5%

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

   7. Applied rewrites 46.7%

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

   if 1.9999999999999999e-94 < (*.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))))) < 2.0000000000000001e299

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-U\right) \cdot \left(-1 + -2 \cdot \left(0.5 \cdot \cos K\right)\right)\right) \cdot \frac{J}{U}, \frac{J}{U}, U\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-U\right) \cdot \left(-1 + -2 \cdot \left(0.5 \cdot \cos K\right)\right)\right) \cdot \frac{J}{U}, \frac{J}{U}, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-245}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -5e-245)
       (* (* -2.0 J) (sqrt (+ 1.0 (pow (/ U_m (* J 2.0)) 2.0))))
       (if (<= t_1 1e+304) (* (* -2.0 J) (cos (* K 0.5))) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e-245) {
		tmp = (-2.0 * J) * sqrt((1.0 + pow((U_m / (J * 2.0)), 2.0)));
	} else if (t_1 <= 1e+304) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= -5e-245) {
		tmp = (-2.0 * J) * Math.sqrt((1.0 + Math.pow((U_m / (J * 2.0)), 2.0)));
	} else if (t_1 <= 1e+304) {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= -5e-245:
		tmp = (-2.0 * J) * math.sqrt((1.0 + math.pow((U_m / (J * 2.0)), 2.0)))
	elif t_1 <= 1e+304:
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-245)
		tmp = Float64(Float64(-2.0 * J) * sqrt(Float64(1.0 + (Float64(U_m / Float64(J * 2.0)) ^ 2.0))));
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= -5e-245)
		tmp = (-2.0 * J) * sqrt((1.0 + ((U_m / (J * 2.0)) ^ 2.0)));
	elseif (t_1 <= 1e+304)
		tmp = (-2.0 * J) * cos((K * 0.5));
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -5e-245], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

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))))) < -inf.0

   1. Initial program 6.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999997e-245

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -4.9999999999999997e-245 < (*.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))))) < 9.9999999999999994e303

   1. Initial program 99.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 66.2

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

   5. Applied rewrites 66.2%

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

   if 9.9999999999999994e303 < (*.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 5.4%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 71.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.2

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

   10. Applied rewrites 71.2%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-245}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-2 \cdot J}{U\_m}, J, -U\_m\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+258}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-154}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (fma (/ (* -2.0 J) U_m) J (- U_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0))))))
   (if (<= t_2 -5e+258)
     t_0
     (if (<= t_2 -4e-154)
       (* (* -2.0 J) (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
       (if (<= t_2 -1e-306) t_0 U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = fma(((-2.0 * J) / U_m), J, -U_m);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -5e+258) {
		tmp = t_0;
	} else if (t_2 <= -4e-154) {
		tmp = (-2.0 * J) * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else if (t_2 <= -1e-306) {
		tmp = t_0;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = fma(Float64(Float64(-2.0 * J) / U_m), J, Float64(-U_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -5e+258)
		tmp = t_0;
	elseif (t_2 <= -4e-154)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J * J)), 1.0)));
	elseif (t_2 <= -1e-306)
		tmp = t_0;
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(N[(-2.0 * J), $MachinePrecision] / U$95$m), $MachinePrecision] * J + (-U$95$m)), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+258], t$95$0, If[LessEqual[t$95$2, -4e-154], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-306], t$95$0, U$95$m]]]]]]

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))))) < -5e258 or -3.9999999999999999e-154 < (*.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))))) < -1.00000000000000003e-306

   1. Initial program 32.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 35.0

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

   5. Applied rewrites 35.0%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 38.1%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 37.1

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

   10. Applied rewrites 37.1%

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

   if -5e258 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.9999999999999999e-154

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 41.7

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

   5. Applied rewrites 41.7%

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

   if -1.00000000000000003e-306 < (*.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 77.6%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 29.6

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 32.4%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 32.4

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

   10. Applied rewrites 32.4%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+258}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 \cdot J}{U}, J, -U\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -4 \cdot 10^{-154}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 \cdot J}{U}, J, -U\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-104}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 \cdot J}{U\_m}, J, -U\_m\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-104)
       (* -2.0 J)
       (if (<= t_1 -1e-306) (fma (/ (* -2.0 J) U_m) J (- U_m)) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-104) {
		tmp = -2.0 * J;
	} else if (t_1 <= -1e-306) {
		tmp = fma(((-2.0 * J) / U_m), J, -U_m);
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-104)
		tmp = Float64(-2.0 * J);
	elseif (t_1 <= -1e-306)
		tmp = fma(Float64(Float64(-2.0 * J) / U_m), J, Float64(-U_m));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-104], N[(-2.0 * J), $MachinePrecision], If[LessEqual[t$95$1, -1e-306], N[(N[(N[(-2.0 * J), $MachinePrecision] / U$95$m), $MachinePrecision] * J + (-U$95$m)), $MachinePrecision], U$95$m]]]]]

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))))) < -inf.0

   1. Initial program 6.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999985e-104

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 47.7

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 34.4

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

   8. Applied rewrites 34.4%

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

   if -1.99999999999999985e-104 < (*.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))))) < -1.00000000000000003e-306

   1. Initial program 99.8%

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

   3. Taylor expanded in 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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 10.8

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

   5. Applied rewrites 10.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 16.2%

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

   8. Taylor expanded in K around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 11.6

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

   10. Applied rewrites 11.6%

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

   if -1.00000000000000003e-306 < (*.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 77.6%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 29.6

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 32.4%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 32.4

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

   10. Applied rewrites 32.4%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-104}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 \cdot J}{U}, J, -U\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-104}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-104) (* -2.0 J) (if (<= t_1 -1e-306) (- U_m) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-104) {
		tmp = -2.0 * J;
	} else if (t_1 <= -1e-306) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= -2e-104) {
		tmp = -2.0 * J;
	} else if (t_1 <= -1e-306) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= -2e-104:
		tmp = -2.0 * J
	elif t_1 <= -1e-306:
		tmp = -U_m
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-104)
		tmp = Float64(-2.0 * J);
	elseif (t_1 <= -1e-306)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= -2e-104)
		tmp = -2.0 * J;
	elseif (t_1 <= -1e-306)
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-104], N[(-2.0 * J), $MachinePrecision], If[LessEqual[t$95$1, -1e-306], (-U$95$m), U$95$m]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or -1.99999999999999985e-104 < (*.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))))) < -1.00000000000000003e-306

   1. Initial program 27.5%

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

   3. Taylor expanded in J around 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 36.8

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

   5. Applied rewrites 36.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999985e-104

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 47.7

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 34.4

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

   8. Applied rewrites 34.4%

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

   if -1.00000000000000003e-306 < (*.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 77.6%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 29.6

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 32.4%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 32.4

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

   10. Applied rewrites 32.4%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-104}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot 2\right)}, \frac{U\_m}{J \cdot 2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e+304)
       (*
        (* -2.0 (* J (cos (* K 0.5))))
        (sqrt
         (fma
          (/ U_m (* (fma 0.5 (cos K) 0.5) (* J 2.0)))
          (/ U_m (* J 2.0))
          1.0)))
       U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+304) {
		tmp = (-2.0 * (J * cos((K * 0.5)))) * sqrt(fma((U_m / (fma(0.5, cos(K), 0.5) * (J * 2.0))), (U_m / (J * 2.0)), 1.0));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(-2.0 * Float64(J * cos(Float64(K * 0.5)))) * sqrt(fma(Float64(U_m / Float64(fma(0.5, cos(K), 0.5) * Float64(J * 2.0))), Float64(U_m / Float64(J * 2.0)), 1.0)));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+304], N[(N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / N[(N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999994e303

   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

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

   8. Applied rewrites 99.6%

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

   if 9.9999999999999994e303 < (*.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 5.4%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 71.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.2

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

   10. Applied rewrites 71.2%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(J \cdot 2\right)}, \frac{U}{J \cdot 2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m}{J \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)} \cdot \frac{U\_m}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e+304)
       (*
        (* -2.0 (* J (cos (* K 0.5))))
        (sqrt
         (fma 0.25 (* (/ U_m (* J (fma 0.5 (cos K) 0.5))) (/ U_m J)) 1.0)))
       U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+304) {
		tmp = (-2.0 * (J * cos((K * 0.5)))) * sqrt(fma(0.25, ((U_m / (J * fma(0.5, cos(K), 0.5))) * (U_m / J)), 1.0));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(-2.0 * Float64(J * cos(Float64(K * 0.5)))) * sqrt(fma(0.25, Float64(Float64(U_m / Float64(J * fma(0.5, cos(K), 0.5))) * Float64(U_m / J)), 1.0)));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+304], N[(N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m / N[(J * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999994e303

   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

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 94.3%

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

   7. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 76.8

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

   9. Applied rewrites 76.8%

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

       1. lift-cos.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 99.1

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

   11. Applied rewrites 99.1%

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

   if 9.9999999999999994e303 < (*.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 5.4%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 71.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.2

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

   10. Applied rewrites 71.2%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U}{J \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)} \cdot \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e+304) (* (* -2.0 J) (cos (* K 0.5))) U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+304) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 1e+304) {
		tmp = (-2.0 * J) * Math.cos((K * 0.5));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 1e+304:
		tmp = (-2.0 * J) * math.cos((K * 0.5))
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 1e+304)
		tmp = (-2.0 * J) * cos((K * 0.5));
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+304], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 44.4

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

   5. Applied rewrites 44.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999994e303

   1. Initial program 99.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 69.0

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

   5. Applied rewrites 69.0%

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

   if 9.9999999999999994e303 < (*.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 5.4%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 71.2%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.2

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

   10. Applied rewrites 71.2%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;\frac{J \cdot \left(-2 \cdot J\right)}{U\_m} - U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-245}:\\ \;\;\;\;\mathsf{fma}\left(J, 0.25 \cdot \left(K \cdot K\right), -2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J}, 0.25 \cdot \frac{U\_m}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 -5e+307)
     (- (/ (* J (* -2.0 J)) U_m) U_m)
     (if (<= t_1 -5e-245)
       (*
        (fma J (* 0.25 (* K K)) (* -2.0 J))
        (sqrt (fma (/ U_m J) (* 0.25 (/ U_m J)) 1.0)))
       U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = ((J * (-2.0 * J)) / U_m) - U_m;
	} else if (t_1 <= -5e-245) {
		tmp = fma(J, (0.25 * (K * K)), (-2.0 * J)) * sqrt(fma((U_m / J), (0.25 * (U_m / J)), 1.0));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+307)
		tmp = Float64(Float64(Float64(J * Float64(-2.0 * J)) / U_m) - U_m);
	elseif (t_1 <= -5e-245)
		tmp = Float64(fma(J, Float64(0.25 * Float64(K * K)), Float64(-2.0 * J)) * sqrt(fma(Float64(U_m / J), Float64(0.25 * Float64(U_m / J)), 1.0)));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+307], N[(N[(N[(J * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-245], N[(N[(J * N[(0.25 * N[(K * K), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / J), $MachinePrecision] * N[(0.25 * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]

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))))) < -5e307

   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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in K around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 43.1

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

   8. Applied rewrites 43.1%

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

   if -5e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999997e-245

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 34.2

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

   8. Applied rewrites 34.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 44.3

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

   10. Applied rewrites 44.3%

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

   if -4.9999999999999997e-245 < (*.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 78.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 29.2

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

   5. Applied rewrites 29.2%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 32.1%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 32.1

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

   10. Applied rewrites 32.1%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+307}:\\ \;\;\;\;\frac{J \cdot \left(-2 \cdot J\right)}{U} - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-245}:\\ \;\;\;\;\mathsf{fma}\left(J, 0.25 \cdot \left(K \cdot K\right), -2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J}, 0.25 \cdot \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.8% accurate, 1.0× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<=
        (*
         (* (* -2.0 J) t_0)
         (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))
        -1e-306)
     (- U_m)
     U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)))) <= -1e-306) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    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 (((((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u_m / (t_0 * (j * 2.0d0))) ** 2.0d0)))) <= (-1d-306)) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))) <= -1e-306) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))) <= -1e-306:
		tmp = -U_m
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0)))) <= -1e-306)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)))) <= -1e-306)
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-306], (-U$95$m), U$95$m]]

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))))) < -1.00000000000000003e-306

   1. Initial program 74.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 21.3

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

   5. Applied rewrites 21.3%

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

   if -1.00000000000000003e-306 < (*.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 77.6%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 29.6

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 32.4%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \cdot -1 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \left(\mathsf{neg}\left(U\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 32.4

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

   10. Applied rewrites 32.4%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.0% accurate, 124.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ -U\_m \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (- U_m))

C

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

Fortran

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

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	return -U_m;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	return -U_m

Julia

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

MATLAB

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

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := (-U$95$m)

Derivation

1. Initial program 75.9%

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

3. Taylor expanded in 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 20.9

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

5. Applied rewrites 20.9%

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

Reproduce

herbie shell --seed 2024219 
(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)))))