Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.1% → 99.7%

Time: 12.6s

Alternatives: 10

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: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ (pow (/ U_m (* t_0 (* J 2.0))) 2.0) 1.0))))
        (t_2 (cos (* K 0.5))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 2e+307)
       (*
        (* -2.0 (* J t_2))
        (sqrt (+ (pow (/ U_m (* (* J 2.0) t_2)) 2.0) 1.0)))
       U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((pow((U_m / (t_0 * (J * 2.0))), 2.0) + 1.0));
	double t_2 = cos((K * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e+307) {
		tmp = (-2.0 * (J * t_2)) * sqrt((pow((U_m / ((J * 2.0) * t_2)), 2.0) + 1.0));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((Math.pow((U_m / (t_0 * (J * 2.0))), 2.0) + 1.0));
	double t_2 = Math.cos((K * 0.5));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 2e+307) {
		tmp = (-2.0 * (J * t_2)) * Math.sqrt((Math.pow((U_m / ((J * 2.0) * t_2)), 2.0) + 1.0));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((math.pow((U_m / (t_0 * (J * 2.0))), 2.0) + 1.0))
	t_2 = math.cos((K * 0.5))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 2e+307:
		tmp = (-2.0 * (J * t_2)) * math.sqrt((math.pow((U_m / ((J * 2.0) * t_2)), 2.0) + 1.0))
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64((Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0) + 1.0)))
	t_2 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e+307)
		tmp = Float64(Float64(-2.0 * Float64(J * t_2)) * sqrt(Float64((Float64(U_m / Float64(Float64(J * 2.0) * t_2)) ^ 2.0) + 1.0)));
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((((U_m / (t_0 * (J * 2.0))) ^ 2.0) + 1.0));
	t_2 = cos((K * 0.5));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 2e+307)
		tmp = (-2.0 * (J * t_2)) * sqrt((((U_m / ((J * 2.0) * t_2)) ^ 2.0) + 1.0));
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $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+307], N[(N[(-2.0 * N[(J * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(J * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 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 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 100.0

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

   5. Simplified 100.0%

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

   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))))) < 1.99999999999999997e307

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   if 1.99999999999999997e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.7%

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

   3. Taylor expanded in 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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in J around 0

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

   8. Simplified 100.0%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{{\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2} + 1}\\ t_2 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+158}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot t\_2\right) \cdot \sqrt{\mathsf{fma}\left(U\_m \cdot 0.25, \frac{U\_m}{J \cdot J}, 1\right)}\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-119}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot t\_2\right) \cdot \sqrt{\mathsf{fma}\left(U\_m \cdot U\_m, \frac{0.25}{\left(J \cdot J\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, 1\right)}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ (pow (/ U_m (* t_0 (* J 2.0))) 2.0) 1.0))))
        (t_2 (cos (* K 0.5))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -5e+158)
       (* -2.0 (* (* J t_2) (sqrt (fma (* U_m 0.25) (/ U_m (* J J)) 1.0))))
       (if (<= t_1 -2e-119)
         (*
          J
          (*
           (* -2.0 t_2)
           (sqrt
            (fma (* U_m U_m) (/ 0.25 (* (* J J) (fma 0.5 (cos K) 0.5))) 1.0))))
         (if (<= t_1 2e+307) (* (* -2.0 J) t_2) U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((pow((U_m / (t_0 * (J * 2.0))), 2.0) + 1.0));
	double t_2 = cos((K * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e+158) {
		tmp = -2.0 * ((J * t_2) * sqrt(fma((U_m * 0.25), (U_m / (J * J)), 1.0)));
	} else if (t_1 <= -2e-119) {
		tmp = J * ((-2.0 * t_2) * sqrt(fma((U_m * U_m), (0.25 / ((J * J) * fma(0.5, cos(K), 0.5))), 1.0)));
	} else if (t_1 <= 2e+307) {
		tmp = (-2.0 * J) * t_2;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64((Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0) + 1.0)))
	t_2 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e+158)
		tmp = Float64(-2.0 * Float64(Float64(J * t_2) * sqrt(fma(Float64(U_m * 0.25), Float64(U_m / Float64(J * J)), 1.0))));
	elseif (t_1 <= -2e-119)
		tmp = Float64(J * Float64(Float64(-2.0 * t_2) * sqrt(fma(Float64(U_m * U_m), Float64(0.25 / Float64(Float64(J * J) * fma(0.5, cos(K), 0.5))), 1.0))));
	elseif (t_1 <= 2e+307)
		tmp = Float64(Float64(-2.0 * J) * t_2);
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $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, -5e+158], N[(-2.0 * N[(N[(J * t$95$2), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m * 0.25), $MachinePrecision] * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-119], N[(J * N[(N[(-2.0 * t$95$2), $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]), $MachinePrecision], If[LessEqual[t$95$1, 2e+307], N[(N[(-2.0 * J), $MachinePrecision] * t$95$2), $MachinePrecision], U$95$m]]]]]]]

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 5.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.7

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

   5. Simplified 99.7%

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

   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))))) < -4.9999999999999996e158

   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 66.1

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

   5. Simplified 66.1%

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

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

   7. Applied egg-rr 84.1%

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

   if -4.9999999999999996e158 < (*.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))))) < -2.00000000000000003e-119

   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

   4. Applied egg-rr 94.4%

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

   5. Taylor expanded in K around inf

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

   7. Simplified 93.5%

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

   if -2.00000000000000003e-119 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.99999999999999997e307

   1. Initial program 99.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 67.6

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

   5. Simplified 67.6%

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

   if 1.99999999999999997e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 6.5%

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

   3. Taylor expanded in U around -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 99.1

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

   5. Simplified 99.1%

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

   6. Taylor expanded in J around 0

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

   8. Simplified 99.2%

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

      1. lift-neg.f64 N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 99.2

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

   10. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{-\left(-U\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{{\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2} + 1} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{{\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2} + 1} \leq -5 \cdot 10^{+158}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(U \cdot 0.25, \frac{U}{J \cdot J}, 1\right)}\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{{\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2} + 1} \leq -2 \cdot 10^{-119}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(U \cdot U, \frac{0.25}{\left(J \cdot J\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, 1\right)}\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{{\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2} + 1} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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)))))