Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.0% → 99.5%

Time: 11.2s

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

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))
        (t_2 (* (* t_0 (* -2.0 J)) t_1)))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 1e+304)
       (* t_1 (* J (* -2.0 (cos (* K 0.5)))))
       (fma 2.0 (/ (* J 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 = sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = (t_0 * (-2.0 * J)) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 * (J * (-2.0 * cos((K * 0.5))));
	} else {
		tmp = fma(2.0, ((J * J) / U_m), 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(t_0 * Float64(-2.0 * J)) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+304)
		tmp = Float64(t_1 * Float64(J * Float64(-2.0 * cos(Float64(K * 0.5)))));
	else
		tmp = fma(2.0, Float64(Float64(J * 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[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[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   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. Simplified 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 K around 0

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

   8. Simplified 71.2%

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

   9. Taylor expanded in J around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 71.2

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

   11. Simplified 71.2%

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

4. Final simplification 89.0%

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

Alternative 2: 83.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(U\_m \cdot U\_m, \frac{0.25}{\left(J \cdot J\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, 1\right)}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(t\_1 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_3 := \frac{U\_m}{-2 \cdot J}\\ t_4 := \left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(t\_3, t\_3, 1\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U\_m} \cdot \left(U\_m \cdot \left(\left(--1\right) - -2 \cdot \left(0.5 \cdot \cos K\right)\right)\right), \frac{J}{U\_m}, U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(U\_m, \frac{\left(J \cdot 2\right) \cdot \frac{J}{U\_m}}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0
         (*
          (* J (* -2.0 (cos (* K 0.5))))
          (sqrt
           (fma (* U_m U_m) (/ 0.25 (* (* J J) (fma 0.5 (cos K) 0.5))) 1.0))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* t_1 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0)))))
        (t_3 (/ U_m (* -2.0 J)))
        (t_4 (* (* -2.0 J) (sqrt (fma t_3 t_3 1.0)))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -5e+202)
       t_4
       (if (<= t_2 -2e-21)
         t_0
         (if (<= t_2 -1e-306)
           t_4
           (if (<= t_2 2e-94)
             (fma
              (* (/ J U_m) (* U_m (- (- -1.0) (* -2.0 (* 0.5 (cos K))))))
              (/ J U_m)
              U_m)
             (if (<= t_2 2e+299)
               t_0
               (fma U_m (/ (* (* J 2.0) (/ J U_m)) U_m) U_m)))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = (J * (-2.0 * cos((K * 0.5)))) * sqrt(fma((U_m * U_m), (0.25 / ((J * J) * fma(0.5, cos(K), 0.5))), 1.0));
	double t_1 = cos((K / 2.0));
	double t_2 = (t_1 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double t_3 = U_m / (-2.0 * J);
	double t_4 = (-2.0 * J) * sqrt(fma(t_3, t_3, 1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -5e+202) {
		tmp = t_4;
	} else if (t_2 <= -2e-21) {
		tmp = t_0;
	} else if (t_2 <= -1e-306) {
		tmp = t_4;
	} else if (t_2 <= 2e-94) {
		tmp = fma(((J / U_m) * (U_m * (-(-1.0) - (-2.0 * (0.5 * cos(K)))))), (J / U_m), U_m);
	} else if (t_2 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = fma(U_m, (((J * 2.0) * (J / U_m)) / U_m), U_m);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(Float64(J * Float64(-2.0 * cos(Float64(K * 0.5)))) * sqrt(fma(Float64(U_m * U_m), Float64(0.25 / Float64(Float64(J * J) * fma(0.5, cos(K), 0.5))), 1.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(t_1 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	t_3 = Float64(U_m / Float64(-2.0 * J))
	t_4 = Float64(Float64(-2.0 * J) * sqrt(fma(t_3, t_3, 1.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -5e+202)
		tmp = t_4;
	elseif (t_2 <= -2e-21)
		tmp = t_0;
	elseif (t_2 <= -1e-306)
		tmp = t_4;
	elseif (t_2 <= 2e-94)
		tmp = fma(Float64(Float64(J / U_m) * Float64(U_m * Float64(Float64(-(-1.0)) - Float64(-2.0 * Float64(0.5 * cos(K)))))), Float64(J / U_m), U_m);
	elseif (t_2 <= 2e+299)
		tmp = t_0;
	else
		tmp = fma(U_m, Float64(Float64(Float64(J * 2.0) * Float64(J / U_m)) / 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[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(0.25 / N[(N[(J * J), $MachinePrecision] * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $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[(t$95$1 * N[(-2.0 * J), $MachinePrecision]), $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[(U$95$m / N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(t$95$3 * t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -5e+202], t$95$4, If[LessEqual[t$95$2, -2e-21], t$95$0, If[LessEqual[t$95$2, -1e-306], t$95$4, If[LessEqual[t$95$2, 2e-94], N[(N[(N[(J / U$95$m), $MachinePrecision] * N[(U$95$m * N[((--1.0) - N[(-2.0 * N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e+299], t$95$0, N[(U$95$m * N[(N[(N[(J * 2.0), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 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. Simplified 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 -1.99999999999999982e-21 < (*.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. *-commutative N/A

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

      2. lower-*.f64 54.1

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

   5. Simplified 54.1%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 61.5

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

   8. Simplified 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. frac-2neg N/A

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

      12. distribute-frac-neg N/A

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

      13. sqr-neg N/A

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

      14. lower-fma.f64 N/A

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

   10. Applied egg-rr 61.5%

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

   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))))) < -1.99999999999999982e-21 or 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. 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

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in K around inf

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-cos.f64 86.2

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

   9. Simplified 86.2%

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

   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))))) < 1.9999999999999999e-94

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 6.2

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

   5. Simplified 6.2%

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

   7. Applied egg-rr 13.8%

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

   if 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. Simplified 67.2%

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

   6. Taylor expanded in K around 0

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

   8. Simplified 67.2%

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

   9. Taylor expanded in U around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 67.4

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

   11. Simplified 67.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-/r* N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 67.6

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

   13. Applied egg-rr 67.6%

       \[\leadsto \mathsf{fma}\left(U, \color{blue}{\frac{\left(2 \cdot J\right) \cdot \frac{J}{U}}{U}}, U\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\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(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\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{\mathsf{fma}\left(\frac{U}{-2 \cdot J}, \frac{U}{-2 \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\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^{-21}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(U \cdot U, \frac{0.25}{\left(J \cdot J\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, 1\right)}\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\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{\mathsf{fma}\left(\frac{U}{-2 \cdot J}, \frac{U}{-2 \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\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(\frac{J}{U} \cdot \left(U \cdot \left(\left(--1\right) - -2 \cdot \left(0.5 \cdot \cos K\right)\right)\right), \frac{J}{U}, U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\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(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(U \cdot U, \frac{0.25}{\left(J \cdot J\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(U, \frac{\left(J \cdot 2\right) \cdot \frac{J}{U}}{U}, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{\left(J \cdot J\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, 0.25, 1\right)}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(t\_1 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_3 := \frac{U\_m}{-2 \cdot J}\\ t_4 := \left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(t\_3, t\_3, 1\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-185}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(U\_m, \frac{\left(J \cdot 2\right) \cdot \frac{J}{U\_m}}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0
         (*
          (* J (* -2.0 (cos (* K 0.5))))
          (sqrt
           (fma (/ (* U_m U_m) (* (* J J) (fma 0.5 (cos K) 0.5))) 0.25 1.0))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* t_1 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0)))))
        (t_3 (/ U_m (* -2.0 J)))
        (t_4 (* (* -2.0 J) (sqrt (fma t_3 t_3 1.0)))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -5e+202)
       t_4
       (if (<= t_2 -5e-39)
         t_0
         (if (<= t_2 2e-185)
           t_4
           (if (<= t_2 2e+299)
             t_0
             (fma U_m (/ (* (* J 2.0) (/ J U_m)) U_m) U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = (J * (-2.0 * cos((K * 0.5)))) * sqrt(fma(((U_m * U_m) / ((J * J) * fma(0.5, cos(K), 0.5))), 0.25, 1.0));
	double t_1 = cos((K / 2.0));
	double t_2 = (t_1 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double t_3 = U_m / (-2.0 * J);
	double t_4 = (-2.0 * J) * sqrt(fma(t_3, t_3, 1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -5e+202) {
		tmp = t_4;
	} else if (t_2 <= -5e-39) {
		tmp = t_0;
	} else if (t_2 <= 2e-185) {
		tmp = t_4;
	} else if (t_2 <= 2e+299) {
		tmp = t_0;
	} else {
		tmp = fma(U_m, (((J * 2.0) * (J / U_m)) / U_m), U_m);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(Float64(J * Float64(-2.0 * cos(Float64(K * 0.5)))) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(Float64(J * J) * fma(0.5, cos(K), 0.5))), 0.25, 1.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(t_1 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	t_3 = Float64(U_m / Float64(-2.0 * J))
	t_4 = Float64(Float64(-2.0 * J) * sqrt(fma(t_3, t_3, 1.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -5e+202)
		tmp = t_4;
	elseif (t_2 <= -5e-39)
		tmp = t_0;
	elseif (t_2 <= 2e-185)
		tmp = t_4;
	elseif (t_2 <= 2e+299)
		tmp = t_0;
	else
		tmp = fma(U_m, Float64(Float64(Float64(J * 2.0) * Float64(J / U_m)) / 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[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(N[(J * J), $MachinePrecision] * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(-2.0 * J), $MachinePrecision]), $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[(U$95$m / N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(t$95$3 * t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -5e+202], t$95$4, If[LessEqual[t$95$2, -5e-39], t$95$0, If[LessEqual[t$95$2, 2e-185], t$95$4, If[LessEqual[t$95$2, 2e+299], t$95$0, N[(U$95$m * N[(N[(N[(J * 2.0), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] / 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. Simplified 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. *-commutative N/A

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

      2. lower-*.f64 55.1

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

   5. Simplified 55.1%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 62.4

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

   8. Simplified 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. frac-2neg N/A

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

      12. distribute-frac-neg N/A

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

      13. sqr-neg N/A

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

      14. lower-fma.f64 N/A

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

   10. Applied egg-rr 62.4%

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

   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

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 95.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   8. Applied egg-rr 95.1%

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

   9. Taylor expanded in K around inf

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

       1. lower-sqrt.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-cos.f64 83.7

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

   11. Simplified 83.7%

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

   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. Simplified 67.2%

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

   6. Taylor expanded in K around 0

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

   8. Simplified 67.2%

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

   9. Taylor expanded in U around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 67.4

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

   11. Simplified 67.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-/r* N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 67.6

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

   13. Applied egg-rr 67.6%

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

4. Final simplification 72.5%

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

Alternative 4: 76.2% accurate, 0.2× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* t_0 (* -2.0 J)))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_3 (/ U_m (* -2.0 J))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -1e-306)
       (* (* -2.0 J) (sqrt (fma t_3 t_3 1.0)))
       (if (<= t_2 2e-94)
         (fma
          (* (/ J U_m) (* U_m (- (- -1.0) (* -2.0 (* 0.5 (cos K))))))
          (/ J U_m)
          U_m)
         (if (<= t_2 2e+299)
           (* t_1 (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
           (fma U_m (/ (* (* J 2.0) (/ J U_m)) 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 = t_0 * (-2.0 * J);
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_3 = U_m / (-2.0 * J);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -1e-306) {
		tmp = (-2.0 * J) * sqrt(fma(t_3, t_3, 1.0));
	} else if (t_2 <= 2e-94) {
		tmp = fma(((J / U_m) * (U_m * (-(-1.0) - (-2.0 * (0.5 * cos(K)))))), (J / U_m), U_m);
	} else if (t_2 <= 2e+299) {
		tmp = t_1 * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else {
		tmp = fma(U_m, (((J * 2.0) * (J / U_m)) / U_m), 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(t_0 * Float64(-2.0 * J))
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_3 = Float64(U_m / Float64(-2.0 * J))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -1e-306)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(t_3, t_3, 1.0)));
	elseif (t_2 <= 2e-94)
		tmp = fma(Float64(Float64(J / U_m) * Float64(U_m * Float64(Float64(-(-1.0)) - Float64(-2.0 * Float64(0.5 * cos(K)))))), Float64(J / U_m), U_m);
	elseif (t_2 <= 2e+299)
		tmp = Float64(t_1 * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J * J)), 1.0)));
	else
		tmp = fma(U_m, Float64(Float64(Float64(J * 2.0) * Float64(J / U_m)) / 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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]), $MachinePrecision]}, Block[{t$95$3 = N[(U$95$m / N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -1e-306], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(t$95$3 * t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-94], N[(N[(N[(J / U$95$m), $MachinePrecision] * N[(U$95$m * N[((--1.0) - N[(-2.0 * N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e+299], N[(t$95$1 * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(U$95$m * N[(N[(N[(J * 2.0), $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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. Simplified 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. *-commutative N/A

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

      2. lower-*.f64 46.9

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

   5. Simplified 46.9%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 52.3

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

   8. Simplified 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. frac-2neg N/A

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

      12. distribute-frac-neg N/A

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

      13. sqr-neg N/A

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

      14. lower-fma.f64 N/A

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

   10. Applied egg-rr 52.3%

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

   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))))) < 1.9999999999999999e-94

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 6.2

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

   5. Simplified 6.2%

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

   7. Applied egg-rr 13.8%

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

   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. Simplified 73.7%

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

   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. Simplified 67.2%

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

   6. Taylor expanded in K around 0

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

   8. Simplified 67.2%

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

   9. Taylor expanded in U around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 67.4

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

   11. Simplified 67.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-/r* N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 67.6

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

   13. Applied egg-rr 67.6%

       \[\leadsto \mathsf{fma}\left(U, \color{blue}{\frac{\left(2 \cdot J\right) \cdot \frac{J}{U}}{U}}, U\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 56.6%

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

Alternative 5: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := -2 \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;J \cdot \left(t\_2 \cdot \sqrt{\mathsf{fma}\left(U\_m, \frac{U\_m}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(K \cdot 0.5\right)\right)\right) \cdot \left(\left(J \cdot 2\right) \cdot \left(J \cdot 2\right)\right)}, 1\right)}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(J \cdot t\_2\right) \cdot \sqrt{1 + \frac{\frac{U\_m}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)} \cdot \left(U\_m \cdot 0.5\right)}{J}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{J \cdot J}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (* -2.0 (cos (* K 0.5)))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -1e+135)
       (*
        J
        (*
         t_2
         (sqrt
          (fma
           U_m
           (/
            U_m
            (*
             (+ 0.5 (* 0.5 (cos (* 2.0 (* K 0.5)))))
             (* (* J 2.0) (* J 2.0))))
           1.0))))
       (if (<= t_1 1e+304)
         (*
          (* J t_2)
          (sqrt
           (+
            1.0
            (/
             (* (/ U_m (* (* J 2.0) (fma 0.5 (cos K) 0.5))) (* U_m 0.5))
             J))))
         (fma 2.0 (/ (* J 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 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = -2.0 * cos((K * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e+135) {
		tmp = J * (t_2 * sqrt(fma(U_m, (U_m / ((0.5 + (0.5 * cos((2.0 * (K * 0.5))))) * ((J * 2.0) * (J * 2.0)))), 1.0)));
	} else if (t_1 <= 1e+304) {
		tmp = (J * t_2) * sqrt((1.0 + (((U_m / ((J * 2.0) * fma(0.5, cos(K), 0.5))) * (U_m * 0.5)) / J)));
	} else {
		tmp = fma(2.0, ((J * J) / U_m), 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(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = Float64(-2.0 * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+135)
		tmp = Float64(J * Float64(t_2 * sqrt(fma(U_m, Float64(U_m / Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(K * 0.5))))) * Float64(Float64(J * 2.0) * Float64(J * 2.0)))), 1.0))));
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(J * t_2) * sqrt(Float64(1.0 + Float64(Float64(Float64(U_m / Float64(Float64(J * 2.0) * fma(0.5, cos(K), 0.5))) * Float64(U_m * 0.5)) / J))));
	else
		tmp = fma(2.0, Float64(Float64(J * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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]}, Block[{t$95$2 = N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e+135], N[(J * N[(t$95$2 * N[Sqrt[N[(U$95$m * N[(U$95$m / N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(K * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(J * 2.0), $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[(J * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(N[(U$95$m / N[(N[(J * 2.0), $MachinePrecision] * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(J * J), $MachinePrecision] / 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. Simplified 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))))) < -9.99999999999999962e134

   1. Initial program 99.7%

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

   4. Applied egg-rr 96.9%

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

   if -9.99999999999999962e134 < (*.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

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 95.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   8. Applied egg-rr 95.5%

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

   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. Simplified 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 K around 0

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

   8. Simplified 71.2%

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

   9. Taylor expanded in J around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 71.2

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

   11. Simplified 71.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{+135}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(K \cdot 0.5\right)\right)\right) \cdot \left(\left(J \cdot 2\right) \cdot \left(J \cdot 2\right)\right)}, 1\right)}\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+304}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{1 + \frac{\frac{U}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)} \cdot \left(U \cdot 0.5\right)}{J}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{J \cdot J}{U}, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+205}:\\ \;\;\;\;-2 \cdot \left(\sqrt{\mathsf{fma}\left(U\_m, \frac{U\_m}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(K \cdot 0.5\right)\right)\right) \cdot \left(\left(J \cdot 2\right) \cdot \left(J \cdot 2\right)\right)}, 1\right)} \cdot \left(J \cdot t\_2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot t\_2\right)\right) \cdot \sqrt{1 + \frac{\frac{U\_m}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)} \cdot \left(U\_m \cdot 0.5\right)}{J}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{J \cdot J}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (cos (* K 0.5))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e+205)
       (*
        -2.0
        (*
         (sqrt
          (fma
           U_m
           (/
            U_m
            (*
             (+ 0.5 (* 0.5 (cos (* 2.0 (* K 0.5)))))
             (* (* J 2.0) (* J 2.0))))
           1.0))
         (* J t_2)))
       (if (<= t_1 1e+304)
         (*
          (* J (* -2.0 t_2))
          (sqrt
           (+
            1.0
            (/
             (* (/ U_m (* (* J 2.0) (fma 0.5 (cos K) 0.5))) (* U_m 0.5))
             J))))
         (fma 2.0 (/ (* J 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 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = cos((K * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e+205) {
		tmp = -2.0 * (sqrt(fma(U_m, (U_m / ((0.5 + (0.5 * cos((2.0 * (K * 0.5))))) * ((J * 2.0) * (J * 2.0)))), 1.0)) * (J * t_2));
	} else if (t_1 <= 1e+304) {
		tmp = (J * (-2.0 * t_2)) * sqrt((1.0 + (((U_m / ((J * 2.0) * fma(0.5, cos(K), 0.5))) * (U_m * 0.5)) / J)));
	} else {
		tmp = fma(2.0, ((J * J) / U_m), 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(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e+205)
		tmp = Float64(-2.0 * Float64(sqrt(fma(U_m, Float64(U_m / Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(K * 0.5))))) * Float64(Float64(J * 2.0) * Float64(J * 2.0)))), 1.0)) * Float64(J * t_2)));
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(J * Float64(-2.0 * t_2)) * sqrt(Float64(1.0 + Float64(Float64(Float64(U_m / Float64(Float64(J * 2.0) * fma(0.5, cos(K), 0.5))) * Float64(U_m * 0.5)) / J))));
	else
		tmp = fma(2.0, Float64(Float64(J * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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]}, Block[{t$95$2 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e+205], N[(-2.0 * N[(N[Sqrt[N[(U$95$m * N[(U$95$m / N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(K * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(J * 2.0), $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[(J * N[(-2.0 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(N[(U$95$m / N[(N[(J * 2.0), $MachinePrecision] * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(J * J), $MachinePrecision] / 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. Simplified 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))))) < -2.00000000000000003e205

   1. Initial program 99.7%

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

   4. Applied egg-rr 95.4%

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

   if -2.00000000000000003e205 < (*.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

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 95.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   8. Applied egg-rr 95.8%

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

   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. Simplified 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 K around 0

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

   8. Simplified 71.2%

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

   9. Taylor expanded in J around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 71.2

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

   11. Simplified 71.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+205}:\\ \;\;\;\;-2 \cdot \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(K \cdot 0.5\right)\right)\right) \cdot \left(\left(J \cdot 2\right) \cdot \left(J \cdot 2\right)\right)}, 1\right)} \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+304}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{1 + \frac{\frac{U}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)} \cdot \left(U \cdot 0.5\right)}{J}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{J \cdot J}{U}, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := \frac{U\_m}{-2 \cdot J}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-245}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{J \cdot J}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (/ U_m (* -2.0 J))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -5e-245)
       (* (* -2.0 J) (sqrt (fma t_2 t_2 1.0)))
       (if (<= t_1 1e+304)
         (* (cos (* K 0.5)) (* -2.0 J))
         (fma 2.0 (/ (* J 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 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = U_m / (-2.0 * J);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e-245) {
		tmp = (-2.0 * J) * sqrt(fma(t_2, t_2, 1.0));
	} else if (t_1 <= 1e+304) {
		tmp = cos((K * 0.5)) * (-2.0 * J);
	} else {
		tmp = fma(2.0, ((J * J) / U_m), 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(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = Float64(U_m / Float64(-2.0 * J))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-245)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(t_2, t_2, 1.0)));
	elseif (t_1 <= 1e+304)
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J));
	else
		tmp = fma(2.0, Float64(Float64(J * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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]}, Block[{t$95$2 = N[(U$95$m / N[(-2.0 * J), $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[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(J * J), $MachinePrecision] / 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. Simplified 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. *-commutative N/A

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

      2. lower-*.f64 47.5

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

   5. Simplified 47.5%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 53.0

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

   8. Simplified 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. frac-2neg N/A

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

      12. distribute-frac-neg N/A

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

      13. sqr-neg N/A

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

      14. lower-fma.f64 N/A

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

   10. Applied egg-rr 53.0%

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

   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))))) < 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. *-commutative N/A

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

      7. lower-*.f64 66.2

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

   5. Simplified 66.2%

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

   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. Simplified 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 K around 0

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

   8. Simplified 71.2%

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

   9. Taylor expanded in J around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 71.2

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

   11. Simplified 71.2%

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

4. Final simplification 59.4%

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

Alternative 8: 56.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+276}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-154}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{J \cdot J}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 -2e+276)
     (- U_m)
     (if (<= t_1 -4e-154)
       (* (* -2.0 J) (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
       (if (<= t_1 -1e-306) (- U_m) (fma 2.0 (/ (* J 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 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -2e+276) {
		tmp = -U_m;
	} else if (t_1 <= -4e-154) {
		tmp = (-2.0 * J) * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else if (t_1 <= -1e-306) {
		tmp = -U_m;
	} else {
		tmp = fma(2.0, ((J * J) / U_m), 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(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -2e+276)
		tmp = Float64(-U_m);
	elseif (t_1 <= -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_1 <= -1e-306)
		tmp = Float64(-U_m);
	else
		tmp = fma(2.0, Float64(Float64(J * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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, -2e+276], (-U$95$m), If[LessEqual[t$95$1, -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$1, -1e-306], (-U$95$m), N[(2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e276 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 25.7%

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

   3. Taylor expanded in J around 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 37.4

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

   5. Simplified 37.4%

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

   if -2.0000000000000001e276 < (*.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. *-commutative N/A

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

      13. lower-*.f64 40.9

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

   5. Simplified 40.9%

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

   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. Simplified 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 K around 0

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

   8. Simplified 29.6%

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

   9. Taylor expanded in J around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 31.1

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

   11. Simplified 31.1%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+276}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -4 \cdot 10^{-154}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{J \cdot J}{U}, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.9% accurate, 0.3× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (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) (fma 2.0 (/ (* J 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 = (t_0 * (-2.0 * J)) * 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 = fma(2.0, ((J * J) / U_m), 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(t_0 * Float64(-2.0 * J)) * 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 = fma(2.0, Float64(Float64(J * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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), N[(2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -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. Simplified 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. *-commutative N/A

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

      2. lower-*.f64 47.7

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

   5. Simplified 47.7%

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

   6. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 34.4

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

   8. Simplified 34.4%

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

   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. 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. Simplified 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 K around 0

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

   8. Simplified 29.6%

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

   9. Taylor expanded in J around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 31.1

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

   11. Simplified 31.1%

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

4. Final simplification 33.1%

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

Alternative 10: 99.4% accurate, 0.4× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* t_0 (* -2.0 J)))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 1e+304)
       (*
        t_1
        (sqrt
         (+
          1.0
          (*
           (/ U_m (* J 2.0))
           (/ U_m (* (* J 2.0) (+ 0.5 (* 0.5 (cos (* 2.0 (* K 0.5)))))))))))
       (fma 2.0 (/ (* J 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 = t_0 * (-2.0 * J);
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 * sqrt((1.0 + ((U_m / (J * 2.0)) * (U_m / ((J * 2.0) * (0.5 + (0.5 * cos((2.0 * (K * 0.5))))))))));
	} else {
		tmp = fma(2.0, ((J * J) / U_m), 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(t_0 * Float64(-2.0 * J))
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+304)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + Float64(Float64(U_m / Float64(J * 2.0)) * Float64(U_m / Float64(Float64(J * 2.0) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(K * 0.5)))))))))));
	else
		tmp = fma(2.0, Float64(Float64(J * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+304], N[(t$95$1 * N[Sqrt[N[(1.0 + N[(N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * N[(U$95$m / N[(N[(J * 2.0), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(K * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

   4. Applied egg-rr 99.6%

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

   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. Simplified 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 K around 0

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

   8. Simplified 71.2%

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

   9. Taylor expanded in J around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 71.2

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

   11. Simplified 71.2%

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

4. Final simplification 88.9%

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

Alternative 11: 96.5% accurate, 0.4× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e+304)
       (*
        (* J (* -2.0 (cos (* K 0.5))))
        (sqrt
         (+
          1.0
          (/ (* (/ U_m (* (* J 2.0) (fma 0.5 (cos K) 0.5))) (* U_m 0.5)) J))))
       (fma 2.0 (/ (* J 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 = (t_0 * (-2.0 * J)) * 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 = (J * (-2.0 * cos((K * 0.5)))) * sqrt((1.0 + (((U_m / ((J * 2.0) * fma(0.5, cos(K), 0.5))) * (U_m * 0.5)) / J)));
	} else {
		tmp = fma(2.0, ((J * J) / U_m), 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(t_0 * Float64(-2.0 * J)) * 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(J * Float64(-2.0 * cos(Float64(K * 0.5)))) * sqrt(Float64(1.0 + Float64(Float64(Float64(U_m / Float64(Float64(J * 2.0) * fma(0.5, cos(K), 0.5))) * Float64(U_m * 0.5)) / J))));
	else
		tmp = fma(2.0, Float64(Float64(J * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(N[(U$95$m / N[(N[(J * 2.0), $MachinePrecision] * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   8. Applied egg-rr 94.3%

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

   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. Simplified 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 K around 0

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

   8. Simplified 71.2%

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

   9. Taylor expanded in J around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 71.2

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

   11. Simplified 71.2%

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

4. Final simplification 84.9%

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

Alternative 12: 61.7% accurate, 0.5× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (/ U_m (* -2.0 J))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -1e-306)
       (* (* -2.0 J) (sqrt (fma t_2 t_2 1.0)))
       (fma 2.0 (/ (* J 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 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = U_m / (-2.0 * J);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e-306) {
		tmp = (-2.0 * J) * sqrt(fma(t_2, t_2, 1.0));
	} else {
		tmp = fma(2.0, ((J * J) / U_m), 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(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = Float64(U_m / Float64(-2.0 * J))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-306)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(t_2, t_2, 1.0)));
	else
		tmp = fma(2.0, Float64(Float64(J * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $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]}, Block[{t$95$2 = N[(U$95$m / N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e-306], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + U$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -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. Simplified 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. *-commutative N/A

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

      2. lower-*.f64 46.9

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

   5. Simplified 46.9%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 52.3

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

   8. Simplified 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. distribute-frac-neg N/A

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

      11. frac-2neg N/A

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

      12. distribute-frac-neg N/A

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

      13. sqr-neg N/A

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

      14. lower-fma.f64 N/A

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

   10. Applied egg-rr 52.3%

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

   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. Simplified 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 K around 0

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

   8. Simplified 29.6%

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

   9. Taylor expanded in J around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 31.1

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

   11. Simplified 31.1%

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

4. Final simplification 40.4%

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

Alternative 13: 40.9% accurate, 31.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 270000000000:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 270000000000.0) (* -2.0 J) (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 270000000000.0) {
		tmp = -2.0 * J;
	} 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) :: tmp
    if (u_m <= 270000000000.0d0) then
        tmp = (-2.0d0) * j
    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 tmp;
	if (U_m <= 270000000000.0) {
		tmp = -2.0 * J;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 270000000000.0:
		tmp = -2.0 * J
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 270000000000.0)
		tmp = Float64(-2.0 * J);
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (U_m <= 270000000000.0)
		tmp = -2.0 * J;
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[U$95$m, 270000000000.0], N[(-2.0 * J), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 2.7e11

   1. Initial program 84.7%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 44.6

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

   5. Simplified 44.6%

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

   6. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.4

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

   8. Simplified 31.4%

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

   if 2.7e11 < U

   1. Initial program 49.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 33.0

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

   5. Simplified 33.0%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 270000000000:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
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. Simplified 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)))))