Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 99.5%

Time: 10.8s

Alternatives: 15

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: 73.9% 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

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+305}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0))
          (* t_0 (* J -2.0)))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e+305) t_1 (* -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 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+305) {
		tmp = t_1;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.sqrt((Math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 1e+305) {
		tmp = t_1;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.sqrt((math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 1e+305:
		tmp = t_1
	else:
		tmp = -1.0 * -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(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+305)
		tmp = t_1;
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = sqrt((((U_m / ((2.0 * J) * t_0)) ^ 2.0) + 1.0)) * (t_0 * (J * -2.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 1e+305)
		tmp = t_1;
	else
		tmp = -1.0 * -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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+305], t$95$1, N[(-1.0 * (-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 5.0%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.9

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

   5. Applied rewrites 55.9%

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

   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))))) < 9.9999999999999994e304

   1. Initial program 99.8%

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

   if 9.9999999999999994e304 < (*.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 6.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 53.6%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+305}:\\ \;\;\;\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(\frac{0.25}{\left(\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J\right) \cdot J} \cdot U\_m, U\_m, 1\right)} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2} + 1} \cdot \left(t\_1 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U\_m}, J \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 10^{-167}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{U\_m}{J} \cdot U\_m}{J}, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+305}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt
           (fma (* (/ 0.25 (* (* (fma (cos K) 0.5 0.5) J) J)) U_m) U_m 1.0))
          (* (cos (* 0.5 K)) (* J -2.0))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_1)) 2.0) 1.0))
          (* t_1 (* J -2.0)))))
   (if (<= t_2 -2e+300)
     (fma (/ J U_m) (* J -2.0) (- U_m))
     (if (<= t_2 -5e-64)
       t_0
       (if (<= t_2 1e-167)
         (* (sqrt (fma (/ (* (/ U_m J) U_m) J) 0.25 1.0)) (* J -2.0))
         (if (<= t_2 1e+305) t_0 (* -1.0 (- U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = sqrt(fma(((0.25 / ((fma(cos(K), 0.5, 0.5) * J) * J)) * U_m), U_m, 1.0)) * (cos((0.5 * K)) * (J * -2.0));
	double t_1 = cos((K / 2.0));
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_1)), 2.0) + 1.0)) * (t_1 * (J * -2.0));
	double tmp;
	if (t_2 <= -2e+300) {
		tmp = fma((J / U_m), (J * -2.0), -U_m);
	} else if (t_2 <= -5e-64) {
		tmp = t_0;
	} else if (t_2 <= 1e-167) {
		tmp = sqrt(fma((((U_m / J) * U_m) / J), 0.25, 1.0)) * (J * -2.0);
	} else if (t_2 <= 1e+305) {
		tmp = t_0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(sqrt(fma(Float64(Float64(0.25 / Float64(Float64(fma(cos(K), 0.5, 0.5) * J) * J)) * U_m), U_m, 1.0)) * Float64(cos(Float64(0.5 * K)) * Float64(J * -2.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0) + 1.0)) * Float64(t_1 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_2 <= -2e+300)
		tmp = fma(Float64(J / U_m), Float64(J * -2.0), Float64(-U_m));
	elseif (t_2 <= -5e-64)
		tmp = t_0;
	elseif (t_2 <= 1e-167)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(U_m / J) * U_m) / J), 0.25, 1.0)) * Float64(J * -2.0));
	elseif (t_2 <= 1e+305)
		tmp = t_0;
	else
		tmp = Float64(-1.0 * Float64(-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[Sqrt[N[(N[(N[(0.25 / N[(N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * J), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] * U$95$m + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+300], N[(N[(J / U$95$m), $MachinePrecision] * N[(J * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, -5e-64], t$95$0, If[LessEqual[t$95$2, 1e-167], N[(N[Sqrt[N[(N[(N[(N[(U$95$m / J), $MachinePrecision] * U$95$m), $MachinePrecision] / J), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+305], t$95$0, N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e300

   1. Initial program 12.3%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 57.1%

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

   if -2.0000000000000001e300 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000033e-64 or 1e-167 < (*.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))))) < 9.9999999999999994e304

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 95.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 95.1%

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

   7. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 93.3%

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

   if -5.00000000000000033e-64 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1e-167

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 22.5

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

   9. Applied rewrites 22.5%

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

   11. Applied rewrites 74.6%

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

   if 9.9999999999999994e304 < (*.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 6.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 53.6%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{\left(\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J\right) \cdot J} \cdot U, U, 1\right)} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{-167}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{U}{J} \cdot U}{J}, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+305}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{\left(\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J\right) \cdot J} \cdot U, U, 1\right)} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := t\_0 \cdot \left(J \cdot -2\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot t\_1\\ t_3 := \sqrt{\mathsf{fma}\left(\frac{U\_m}{J \cdot J} \cdot U\_m, 0.25, 1\right)} \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U\_m}, J \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-167}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{U\_m}{J} \cdot U\_m}{J}, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+305}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* t_0 (* J -2.0)))
        (t_2 (* (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0)) t_1))
        (t_3 (* (sqrt (fma (* (/ U_m (* J J)) U_m) 0.25 1.0)) t_1)))
   (if (<= t_2 -2e+300)
     (fma (/ J U_m) (* J -2.0) (- U_m))
     (if (<= t_2 -1e-6)
       t_3
       (if (<= t_2 1e-167)
         (* (sqrt (fma (/ (* (/ U_m J) U_m) J) 0.25 1.0)) (* J -2.0))
         (if (<= t_2 1e+305) t_3 (* -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 = t_0 * (J * -2.0);
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * t_1;
	double t_3 = sqrt(fma(((U_m / (J * J)) * U_m), 0.25, 1.0)) * t_1;
	double tmp;
	if (t_2 <= -2e+300) {
		tmp = fma((J / U_m), (J * -2.0), -U_m);
	} else if (t_2 <= -1e-6) {
		tmp = t_3;
	} else if (t_2 <= 1e-167) {
		tmp = sqrt(fma((((U_m / J) * U_m) / J), 0.25, 1.0)) * (J * -2.0);
	} else if (t_2 <= 1e+305) {
		tmp = t_3;
	} else {
		tmp = -1.0 * -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(J * -2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * t_1)
	t_3 = Float64(sqrt(fma(Float64(Float64(U_m / Float64(J * J)) * U_m), 0.25, 1.0)) * t_1)
	tmp = 0.0
	if (t_2 <= -2e+300)
		tmp = fma(Float64(J / U_m), Float64(J * -2.0), Float64(-U_m));
	elseif (t_2 <= -1e-6)
		tmp = t_3;
	elseif (t_2 <= 1e-167)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(U_m / J) * U_m) / J), 0.25, 1.0)) * Float64(J * -2.0));
	elseif (t_2 <= 1e+305)
		tmp = t_3;
	else
		tmp = Float64(-1.0 * Float64(-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[(J * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+300], N[(N[(J / U$95$m), $MachinePrecision] * N[(J * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, -1e-6], t$95$3, If[LessEqual[t$95$2, 1e-167], N[(N[Sqrt[N[(N[(N[(N[(U$95$m / J), $MachinePrecision] * U$95$m), $MachinePrecision] / J), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+305], t$95$3, N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e300

   1. Initial program 12.3%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 57.1%

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

   if -2.0000000000000001e300 < (*.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))))) < -9.99999999999999955e-7 or 1e-167 < (*.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))))) < 9.9999999999999994e304

   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -9.99999999999999955e-7 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1e-167

   1. Initial program 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 34.4

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

   9. Applied rewrites 34.4%

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

   11. Applied rewrites 74.4%

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

   if 9.9999999999999994e304 < (*.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 6.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 53.6%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{J \cdot J} \cdot U, 0.25, 1\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{-167}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{U}{J} \cdot U}{J}, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+305}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{J \cdot J} \cdot U, 0.25, 1\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2} + 1} \cdot \left(t\_1 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U\_m}, J \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{U\_m}{J} \cdot U\_m}{J}, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+305}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* (cos (* 0.5 K)) (* J -2.0)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_1)) 2.0) 1.0))
          (* t_1 (* J -2.0)))))
   (if (<= t_2 -2e+300)
     (fma (/ J U_m) (* J -2.0) (- U_m))
     (if (<= t_2 -1e-6)
       t_0
       (if (<= t_2 -1e-185)
         (* (sqrt (fma (/ (* (/ U_m J) U_m) J) 0.25 1.0)) (* J -2.0))
         (if (<= t_2 1e+305) t_0 (* -1.0 (- U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((0.5 * K)) * (J * -2.0);
	double t_1 = cos((K / 2.0));
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_1)), 2.0) + 1.0)) * (t_1 * (J * -2.0));
	double tmp;
	if (t_2 <= -2e+300) {
		tmp = fma((J / U_m), (J * -2.0), -U_m);
	} else if (t_2 <= -1e-6) {
		tmp = t_0;
	} else if (t_2 <= -1e-185) {
		tmp = sqrt(fma((((U_m / J) * U_m) / J), 0.25, 1.0)) * (J * -2.0);
	} else if (t_2 <= 1e+305) {
		tmp = t_0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(cos(Float64(0.5 * K)) * Float64(J * -2.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0) + 1.0)) * Float64(t_1 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_2 <= -2e+300)
		tmp = fma(Float64(J / U_m), Float64(J * -2.0), Float64(-U_m));
	elseif (t_2 <= -1e-6)
		tmp = t_0;
	elseif (t_2 <= -1e-185)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(U_m / J) * U_m) / J), 0.25, 1.0)) * Float64(J * -2.0));
	elseif (t_2 <= 1e+305)
		tmp = t_0;
	else
		tmp = Float64(-1.0 * Float64(-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[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+300], N[(N[(J / U$95$m), $MachinePrecision] * N[(J * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, -1e-6], t$95$0, If[LessEqual[t$95$2, -1e-185], N[(N[Sqrt[N[(N[(N[(N[(U$95$m / J), $MachinePrecision] * U$95$m), $MachinePrecision] / J), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+305], t$95$0, N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e300

   1. Initial program 12.3%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 57.1%

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

   if -2.0000000000000001e300 < (*.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))))) < -9.99999999999999955e-7 or -9.9999999999999999e-186 < (*.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))))) < 9.9999999999999994e304

   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 U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -9.99999999999999955e-7 < (*.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))))) < -9.9999999999999999e-186

   1. Initial program 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.2

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

   9. Applied rewrites 49.2%

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

   11. Applied rewrites 81.8%

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

   if 9.9999999999999994e304 < (*.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 6.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 53.6%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{U}{J} \cdot U}{J}, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+305}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2} + 1} \cdot \left(t\_1 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+221}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{U\_m}{\left(\cos K + 1\right) \cdot J} \cdot 0.5}{J}, U\_m, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+305}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.5 \cdot U\_m}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J} \cdot U\_m, \frac{0.5}{J}, 1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* (cos (* 0.5 K)) (* J -2.0)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_1)) 2.0) 1.0))
          (* t_1 (* J -2.0)))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -2e+221)
       (* t_0 (sqrt (fma (/ (* (/ U_m (* (+ (cos K) 1.0) J)) 0.5) J) U_m 1.0)))
       (if (<= t_2 1e+305)
         (*
          (sqrt
           (fma
            (* (/ (* 0.5 U_m) (* (fma (cos K) 0.5 0.5) J)) U_m)
            (/ 0.5 J)
            1.0))
          t_0)
         (* -1.0 (- U_m)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((0.5 * K)) * (J * -2.0);
	double t_1 = cos((K / 2.0));
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_1)), 2.0) + 1.0)) * (t_1 * (J * -2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -2e+221) {
		tmp = t_0 * sqrt(fma((((U_m / ((cos(K) + 1.0) * J)) * 0.5) / J), U_m, 1.0));
	} else if (t_2 <= 1e+305) {
		tmp = sqrt(fma((((0.5 * U_m) / (fma(cos(K), 0.5, 0.5) * J)) * U_m), (0.5 / J), 1.0)) * t_0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(cos(Float64(0.5 * K)) * Float64(J * -2.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0) + 1.0)) * Float64(t_1 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -2e+221)
		tmp = Float64(t_0 * sqrt(fma(Float64(Float64(Float64(U_m / Float64(Float64(cos(K) + 1.0) * J)) * 0.5) / J), U_m, 1.0)));
	elseif (t_2 <= 1e+305)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(0.5 * U_m) / Float64(fma(cos(K), 0.5, 0.5) * J)) * U_m), Float64(0.5 / J), 1.0)) * t_0);
	else
		tmp = Float64(-1.0 * Float64(-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[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -2e+221], N[(t$95$0 * N[Sqrt[N[(N[(N[(N[(U$95$m / N[(N[(N[Cos[K], $MachinePrecision] + 1.0), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / J), $MachinePrecision] * U$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+305], N[(N[Sqrt[N[(N[(N[(N[(0.5 * U$95$m), $MachinePrecision] / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] * N[(0.5 / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(-1.0 * (-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 5.0%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.9

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

   5. Applied rewrites 55.9%

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

   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))))) < -2.0000000000000001e221

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 80.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 80.4%

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

   7. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 88.6%

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

   11. Applied rewrites 99.7%

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

   if -2.0000000000000001e221 < (*.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))))) < 9.9999999999999994e304

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 97.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 97.8

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

   8. Applied rewrites 97.8%

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

   if 9.9999999999999994e304 < (*.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 6.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 53.6%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+221}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{U}{\left(\cos K + 1\right) \cdot J} \cdot 0.5}{J}, U, 1\right)}\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+305}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.5 \cdot U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J} \cdot U, \frac{0.5}{J}, 1\right)} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := t\_0 \cdot \left(J \cdot -2\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U\_m}, J \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-167}:\\ \;\;\;\;\sqrt{\frac{\frac{U\_m}{2 \cdot J} \cdot U\_m}{2 \cdot J} + 1} \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+305}:\\ \;\;\;\;\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{\left(\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J\right) \cdot J}, 0.25, 1\right)}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* t_0 (* J -2.0)))
        (t_2 (* (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0)) t_1)))
   (if (<= t_2 -2e+300)
     (fma (/ J U_m) (* J -2.0) (- U_m))
     (if (<= t_2 1e-167)
       (* (sqrt (+ (/ (* (/ U_m (* 2.0 J)) U_m) (* 2.0 J)) 1.0)) t_1)
       (if (<= t_2 1e+305)
         (*
          (*
           (* (cos (* 0.5 K)) J)
           (sqrt
            (fma (/ (* U_m U_m) (* (* (fma (cos K) 0.5 0.5) J) J)) 0.25 1.0)))
          -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 = t_0 * (J * -2.0);
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * t_1;
	double tmp;
	if (t_2 <= -2e+300) {
		tmp = fma((J / U_m), (J * -2.0), -U_m);
	} else if (t_2 <= 1e-167) {
		tmp = sqrt(((((U_m / (2.0 * J)) * U_m) / (2.0 * J)) + 1.0)) * t_1;
	} else if (t_2 <= 1e+305) {
		tmp = ((cos((0.5 * K)) * J) * sqrt(fma(((U_m * U_m) / ((fma(cos(K), 0.5, 0.5) * J) * J)), 0.25, 1.0))) * -2.0;
	} else {
		tmp = -1.0 * -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(J * -2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * t_1)
	tmp = 0.0
	if (t_2 <= -2e+300)
		tmp = fma(Float64(J / U_m), Float64(J * -2.0), Float64(-U_m));
	elseif (t_2 <= 1e-167)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(U_m / Float64(2.0 * J)) * U_m) / Float64(2.0 * J)) + 1.0)) * t_1);
	elseif (t_2 <= 1e+305)
		tmp = Float64(Float64(Float64(cos(Float64(0.5 * K)) * J) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(Float64(fma(cos(K), 0.5, 0.5) * J) * J)), 0.25, 1.0))) * -2.0);
	else
		tmp = Float64(-1.0 * Float64(-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[(J * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+300], N[(N[(J / U$95$m), $MachinePrecision] * N[(J * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, 1e-167], N[(N[Sqrt[N[(N[(N[(N[(U$95$m / N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] / N[(2.0 * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1e+305], N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * J), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e300

   1. Initial program 12.3%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 57.1%

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

   if -2.0000000000000001e300 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1e-167

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.3

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

   7. Applied rewrites 86.3%

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

   if 1e-167 < (*.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))))) < 9.9999999999999994e304

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 84.1%

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

   if 9.9999999999999994e304 < (*.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 6.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 53.6%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{-167}:\\ \;\;\;\;\sqrt{\frac{\frac{U}{2 \cdot J} \cdot U}{2 \cdot J} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+305}:\\ \;\;\;\;\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U \cdot U}{\left(\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J\right) \cdot J}, 0.25, 1\right)}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{J}{U\_m}, J \cdot -2, -U\_m\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2} + 1} \cdot \left(t\_1 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U\_m}{J \cdot J} \cdot U\_m, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-222}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (fma (/ J U_m) (* J -2.0) (- U_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_1)) 2.0) 1.0))
          (* t_1 (* J -2.0)))))
   (if (<= t_2 -2e+300)
     t_0
     (if (<= t_2 -5e-64)
       (* (sqrt (fma (* (/ U_m (* J J)) U_m) 0.25 1.0)) (* J -2.0))
       (if (<= t_2 -4e-222) t_0 (* -1.0 (- U_m)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = fma((J / U_m), (J * -2.0), -U_m);
	double t_1 = cos((K / 2.0));
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_1)), 2.0) + 1.0)) * (t_1 * (J * -2.0));
	double tmp;
	if (t_2 <= -2e+300) {
		tmp = t_0;
	} else if (t_2 <= -5e-64) {
		tmp = sqrt(fma(((U_m / (J * J)) * U_m), 0.25, 1.0)) * (J * -2.0);
	} else if (t_2 <= -4e-222) {
		tmp = t_0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = fma(Float64(J / U_m), Float64(J * -2.0), Float64(-U_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0) + 1.0)) * Float64(t_1 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_2 <= -2e+300)
		tmp = t_0;
	elseif (t_2 <= -5e-64)
		tmp = Float64(sqrt(fma(Float64(Float64(U_m / Float64(J * J)) * U_m), 0.25, 1.0)) * Float64(J * -2.0));
	elseif (t_2 <= -4e-222)
		tmp = t_0;
	else
		tmp = Float64(-1.0 * Float64(-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 / U$95$m), $MachinePrecision] * N[(J * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+300], t$95$0, If[LessEqual[t$95$2, -5e-64], N[(N[Sqrt[N[(N[(N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-222], t$95$0, N[(-1.0 * (-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))))) < -2.0000000000000001e300 or -5.00000000000000033e-64 < (*.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))))) < -4.00000000000000019e-222

   1. Initial program 35.4%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 48.9

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

   5. Applied rewrites 48.9%

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 52.2%

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

   if -2.0000000000000001e300 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000033e-64

   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   if -4.00000000000000019e-222 < (*.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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 22.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U}{J \cdot J} \cdot U, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -4 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{J}{U\_m}, J \cdot -2, -U\_m\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2} + 1} \cdot \left(t\_1 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-58}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-222}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (fma (/ J U_m) (* J -2.0) (- U_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_1)) 2.0) 1.0))
          (* t_1 (* J -2.0)))))
   (if (<= t_2 -2e+300)
     t_0
     (if (<= t_2 -2e-58)
       (* J -2.0)
       (if (<= t_2 -4e-222) t_0 (* -1.0 (- U_m)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = fma((J / U_m), (J * -2.0), -U_m);
	double t_1 = cos((K / 2.0));
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_1)), 2.0) + 1.0)) * (t_1 * (J * -2.0));
	double tmp;
	if (t_2 <= -2e+300) {
		tmp = t_0;
	} else if (t_2 <= -2e-58) {
		tmp = J * -2.0;
	} else if (t_2 <= -4e-222) {
		tmp = t_0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = fma(Float64(J / U_m), Float64(J * -2.0), Float64(-U_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0) + 1.0)) * Float64(t_1 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_2 <= -2e+300)
		tmp = t_0;
	elseif (t_2 <= -2e-58)
		tmp = Float64(J * -2.0);
	elseif (t_2 <= -4e-222)
		tmp = t_0;
	else
		tmp = Float64(-1.0 * Float64(-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 / U$95$m), $MachinePrecision] * N[(J * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+300], t$95$0, If[LessEqual[t$95$2, -2e-58], N[(J * -2.0), $MachinePrecision], If[LessEqual[t$95$2, -4e-222], t$95$0, N[(-1.0 * (-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))))) < -2.0000000000000001e300 or -2.0000000000000001e-58 < (*.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))))) < -4.00000000000000019e-222

   1. Initial program 36.6%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 48.1

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 51.3%

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

   if -2.0000000000000001e300 < (*.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))))) < -2.0000000000000001e-58

   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 U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 44.0%

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

   if -4.00000000000000019e-222 < (*.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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 22.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-58}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -4 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-58}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-222}:\\ \;\;\;\;\frac{J \cdot J}{U\_m} \cdot -2 - U\_m\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0))
          (* t_0 (* J -2.0)))))
   (if (<= t_1 -5e+306)
     (- U_m)
     (if (<= t_1 -2e-58)
       (* J -2.0)
       (if (<= t_1 -4e-222)
         (- (* (/ (* J J) U_m) -2.0) U_m)
         (* -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 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0));
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = -U_m;
	} else if (t_1 <= -2e-58) {
		tmp = J * -2.0;
	} else if (t_1 <= -4e-222) {
		tmp = (((J * J) / U_m) * -2.0) - U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = sqrt((((u_m / ((2.0d0 * j) * t_0)) ** 2.0d0) + 1.0d0)) * (t_0 * (j * (-2.0d0)))
    if (t_1 <= (-5d+306)) then
        tmp = -u_m
    else if (t_1 <= (-2d-58)) then
        tmp = j * (-2.0d0)
    else if (t_1 <= (-4d-222)) then
        tmp = (((j * j) / u_m) * (-2.0d0)) - u_m
    else
        tmp = (-1.0d0) * -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.sqrt((Math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0));
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = -U_m;
	} else if (t_1 <= -2e-58) {
		tmp = J * -2.0;
	} else if (t_1 <= -4e-222) {
		tmp = (((J * J) / U_m) * -2.0) - U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.sqrt((math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0))
	tmp = 0
	if t_1 <= -5e+306:
		tmp = -U_m
	elif t_1 <= -2e-58:
		tmp = J * -2.0
	elif t_1 <= -4e-222:
		tmp = (((J * J) / U_m) * -2.0) - U_m
	else:
		tmp = -1.0 * -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(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_1 <= -5e+306)
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-58)
		tmp = Float64(J * -2.0);
	elseif (t_1 <= -4e-222)
		tmp = Float64(Float64(Float64(Float64(J * J) / U_m) * -2.0) - U_m);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = sqrt((((U_m / ((2.0 * J) * t_0)) ^ 2.0) + 1.0)) * (t_0 * (J * -2.0));
	tmp = 0.0;
	if (t_1 <= -5e+306)
		tmp = -U_m;
	elseif (t_1 <= -2e-58)
		tmp = J * -2.0;
	elseif (t_1 <= -4e-222)
		tmp = (((J * J) / U_m) * -2.0) - U_m;
	else
		tmp = -1.0 * -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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+306], (-U$95$m), If[LessEqual[t$95$1, -2e-58], N[(J * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -4e-222], N[(N[(N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] * -2.0), $MachinePrecision] - U$95$m), $MachinePrecision], N[(-1.0 * (-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))))) < -4.99999999999999993e306

   1. Initial program 7.6%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 57.1

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

   5. Applied rewrites 57.1%

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

   if -4.99999999999999993e306 < (*.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))))) < -2.0000000000000001e-58

   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 U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 44.2%

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

   if -2.0000000000000001e-58 < (*.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))))) < -4.00000000000000019e-222

   1. Initial program 99.8%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 36.2

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

   5. Applied rewrites 36.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 36.2%

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

   if -4.00000000000000019e-222 < (*.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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 22.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{+306}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-58}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -4 \cdot 10^{-222}:\\ \;\;\;\;\frac{J \cdot J}{U} \cdot -2 - U\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+306}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-58}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-222}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0))
          (* t_0 (* J -2.0)))))
   (if (<= t_1 -5e+306)
     (- U_m)
     (if (<= t_1 -2e-58)
       (* J -2.0)
       (if (<= t_1 -4e-222) (- U_m) (* -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 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0));
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = -U_m;
	} else if (t_1 <= -2e-58) {
		tmp = J * -2.0;
	} else if (t_1 <= -4e-222) {
		tmp = -U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = sqrt((((u_m / ((2.0d0 * j) * t_0)) ** 2.0d0) + 1.0d0)) * (t_0 * (j * (-2.0d0)))
    if (t_1 <= (-5d+306)) then
        tmp = -u_m
    else if (t_1 <= (-2d-58)) then
        tmp = j * (-2.0d0)
    else if (t_1 <= (-4d-222)) then
        tmp = -u_m
    else
        tmp = (-1.0d0) * -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.sqrt((Math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0));
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = -U_m;
	} else if (t_1 <= -2e-58) {
		tmp = J * -2.0;
	} else if (t_1 <= -4e-222) {
		tmp = -U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.sqrt((math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0))
	tmp = 0
	if t_1 <= -5e+306:
		tmp = -U_m
	elif t_1 <= -2e-58:
		tmp = J * -2.0
	elif t_1 <= -4e-222:
		tmp = -U_m
	else:
		tmp = -1.0 * -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(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_1 <= -5e+306)
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-58)
		tmp = Float64(J * -2.0);
	elseif (t_1 <= -4e-222)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = sqrt((((U_m / ((2.0 * J) * t_0)) ^ 2.0) + 1.0)) * (t_0 * (J * -2.0));
	tmp = 0.0;
	if (t_1 <= -5e+306)
		tmp = -U_m;
	elseif (t_1 <= -2e-58)
		tmp = J * -2.0;
	elseif (t_1 <= -4e-222)
		tmp = -U_m;
	else
		tmp = -1.0 * -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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+306], (-U$95$m), If[LessEqual[t$95$1, -2e-58], N[(J * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -4e-222], (-U$95$m), N[(-1.0 * (-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))))) < -4.99999999999999993e306 or -2.0000000000000001e-58 < (*.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))))) < -4.00000000000000019e-222

   1. Initial program 34.2%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.1

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

   5. Applied rewrites 51.1%

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

   if -4.99999999999999993e306 < (*.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))))) < -2.0000000000000001e-58

   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 U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 44.2%

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

   if -4.00000000000000019e-222 < (*.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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 22.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{+306}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{-58}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -4 \cdot 10^{-222}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+305}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{U\_m}{\left(\cos K + 1\right) \cdot J} \cdot 0.5}{J}, U\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0))
          (* t_0 (* J -2.0)))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e+305)
       (*
        (* (cos (* 0.5 K)) (* J -2.0))
        (sqrt (fma (/ (* (/ U_m (* (+ (cos K) 1.0) J)) 0.5) J) U_m 1.0)))
       (* -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 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+305) {
		tmp = (cos((0.5 * K)) * (J * -2.0)) * sqrt(fma((((U_m / ((cos(K) + 1.0) * J)) * 0.5) / J), U_m, 1.0));
	} else {
		tmp = -1.0 * -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(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+305)
		tmp = Float64(Float64(cos(Float64(0.5 * K)) * Float64(J * -2.0)) * sqrt(fma(Float64(Float64(Float64(U_m / Float64(Float64(cos(K) + 1.0) * J)) * 0.5) / J), U_m, 1.0)));
	else
		tmp = Float64(-1.0 * Float64(-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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+305], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(U$95$m / N[(N[(N[Cos[K], $MachinePrecision] + 1.0), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / J), $MachinePrecision] * U$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * (-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 5.0%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.9

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

   5. Applied rewrites 55.9%

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

   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))))) < 9.9999999999999994e304

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 94.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.8%

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

   7. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 84.5%

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

   11. Applied rewrites 94.0%

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

   if 9.9999999999999994e304 < (*.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 6.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 53.6%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+305}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{U}{\left(\cos K + 1\right) \cdot J} \cdot 0.5}{J}, U, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := t\_0 \cdot \left(J \cdot -2\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U\_m}, J \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+305}:\\ \;\;\;\;\sqrt{\frac{\frac{U\_m}{2 \cdot J} \cdot U\_m}{2 \cdot J} + 1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* t_0 (* J -2.0)))
        (t_2 (* (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0)) t_1)))
   (if (<= t_2 -2e+300)
     (fma (/ J U_m) (* J -2.0) (- U_m))
     (if (<= t_2 1e+305)
       (* (sqrt (+ (/ (* (/ U_m (* 2.0 J)) U_m) (* 2.0 J)) 1.0)) t_1)
       (* -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 = t_0 * (J * -2.0);
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * t_1;
	double tmp;
	if (t_2 <= -2e+300) {
		tmp = fma((J / U_m), (J * -2.0), -U_m);
	} else if (t_2 <= 1e+305) {
		tmp = sqrt(((((U_m / (2.0 * J)) * U_m) / (2.0 * J)) + 1.0)) * t_1;
	} else {
		tmp = -1.0 * -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(J * -2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * t_1)
	tmp = 0.0
	if (t_2 <= -2e+300)
		tmp = fma(Float64(J / U_m), Float64(J * -2.0), Float64(-U_m));
	elseif (t_2 <= 1e+305)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(U_m / Float64(2.0 * J)) * U_m) / Float64(2.0 * J)) + 1.0)) * t_1);
	else
		tmp = Float64(-1.0 * Float64(-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[(J * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+300], N[(N[(J / U$95$m), $MachinePrecision] * N[(J * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, 1e+305], N[(N[Sqrt[N[(N[(N[(N[(U$95$m / N[(2.0 * J), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] / N[(2.0 * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(-1.0 * (-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))))) < -2.0000000000000001e300

   1. Initial program 12.3%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 57.1%

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

   if -2.0000000000000001e300 < (*.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))))) < 9.9999999999999994e304

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.7

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

   7. Applied rewrites 85.7%

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

   if 9.9999999999999994e304 < (*.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 6.0%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 53.6%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+305}:\\ \;\;\;\;\sqrt{\frac{\frac{U}{2 \cdot J} \cdot U}{2 \cdot J} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J \cdot -2\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U\_m}, J \cdot -2, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{U\_m}{J} \cdot U\_m}{J}, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0))
          (* t_0 (* J -2.0)))))
   (if (<= t_1 -2e+300)
     (fma (/ J U_m) (* J -2.0) (- U_m))
     (if (<= t_1 -4e-222)
       (* (sqrt (fma (/ (* (/ U_m J) U_m) J) 0.25 1.0)) (* J -2.0))
       (* -1.0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * (t_0 * (J * -2.0));
	double tmp;
	if (t_1 <= -2e+300) {
		tmp = fma((J / U_m), (J * -2.0), -U_m);
	} else if (t_1 <= -4e-222) {
		tmp = sqrt(fma((((U_m / J) * U_m) / J), 0.25, 1.0)) * (J * -2.0);
	} else {
		tmp = -1.0 * -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(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J * -2.0)))
	tmp = 0.0
	if (t_1 <= -2e+300)
		tmp = fma(Float64(J / U_m), Float64(J * -2.0), Float64(-U_m));
	elseif (t_1 <= -4e-222)
		tmp = Float64(sqrt(fma(Float64(Float64(Float64(U_m / J) * U_m) / J), 0.25, 1.0)) * Float64(J * -2.0));
	else
		tmp = Float64(-1.0 * Float64(-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[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+300], N[(N[(J / U$95$m), $MachinePrecision] * N[(J * -2.0), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, -4e-222], N[(N[Sqrt[N[(N[(N[(N[(U$95$m / J), $MachinePrecision] * U$95$m), $MachinePrecision] / J), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 * (-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))))) < -2.0000000000000001e300

   1. Initial program 12.3%

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

   3. Taylor expanded in J around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 57.1%

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

   if -2.0000000000000001e300 < (*.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))))) < -4.00000000000000019e-222

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 94.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   6. Applied rewrites 94.5%

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

   7. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 44.2

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

   9. Applied rewrites 44.2%

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

   11. Applied rewrites 57.9%

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

   if -4.00000000000000019e-222 < (*.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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 22.6%

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

   6. Taylor expanded in U around inf

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

   8. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U}, J \cdot -2, -U\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -4 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{\frac{U}{J} \cdot U}{J}, 0.25, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.8% accurate, 31.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 1.8 \cdot 10^{+38}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (if (<= J 1.8e+38) (- U_m) (* J -2.0)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1.8e+38) {
		tmp = -U_m;
	} else {
		tmp = J * -2.0;
	}
	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 (j <= 1.8d+38) then
        tmp = -u_m
    else
        tmp = j * (-2.0d0)
    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 (J <= 1.8e+38) {
		tmp = -U_m;
	} else {
		tmp = J * -2.0;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 1.8e+38:
		tmp = -U_m
	else:
		tmp = J * -2.0
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 1.8e+38)
		tmp = Float64(-U_m);
	else
		tmp = Float64(J * -2.0);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 1.8e+38)
		tmp = -U_m;
	else
		tmp = J * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 1.8e+38], (-U$95$m), N[(J * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.79999999999999985e38

   1. Initial program 68.1%

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

   3. Taylor expanded in U around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.8

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

   5. Applied rewrites 28.8%

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

   if 1.79999999999999985e38 < J

   1. Initial program 98.5%

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

   3. Taylor expanded in U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 47.9%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.8 \cdot 10^{+38}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.2% 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 75.5%

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

3. Taylor expanded in U around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 25.4

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

5. Applied rewrites 25.4%

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

Reproduce

herbie shell --seed 2024235 
(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)))))