Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.4% → 87.2%

Time: 7.9s

Alternatives: 21

Speedup: 0.3×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    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: 73.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    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: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(K \cdot 0.5\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \cos \left(0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(t\_4 \cdot \sqrt{0.25 \cdot \frac{{U\_m}^{2}}{{t\_4}^{2}}}\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot -2\right) \cdot J\_m\right) \cdot \left(\frac{0.5}{\left|t\_0\right| \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (* K 0.5)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0)))))
        (t_4 (cos (* 0.5 K))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (* -2.0 (* t_4 (sqrt (* 0.25 (/ (pow U_m 2.0) (pow t_4 2.0))))))
      (if (<= t_3 1e+290)
        (*
         (* (* -2.0 J_m) t_1)
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))
        (* (* (* t_0 -2.0) J_m) (* (/ 0.5 (* (fabs t_0) J_m)) U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K * 0.5));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double t_4 = cos((0.5 * K));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -2.0 * (t_4 * sqrt((0.25 * (pow(U_m, 2.0) / pow(t_4, 2.0)))));
	} else if (t_3 <= 1e+290) {
		tmp = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	} else {
		tmp = ((t_0 * -2.0) * J_m) * ((0.5 / (fabs(t_0) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.cos((K * 0.5));
	double t_2 = Math.cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double t_4 = Math.cos((0.5 * K));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (t_4 * Math.sqrt((0.25 * (Math.pow(U_m, 2.0) / Math.pow(t_4, 2.0)))));
	} else if (t_3 <= 1e+290) {
		tmp = ((-2.0 * J_m) * t_1) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	} else {
		tmp = ((t_0 * -2.0) * J_m) * ((0.5 / (Math.abs(t_0) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.cos((K * 0.5))
	t_2 = math.cos((K / 2.0))
	t_3 = ((-2.0 * J_m) * t_2) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_2)), 2.0)))
	t_4 = math.cos((0.5 * K))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = -2.0 * (t_4 * math.sqrt((0.25 * (math.pow(U_m, 2.0) / math.pow(t_4, 2.0)))))
	elif t_3 <= 1e+290:
		tmp = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	else:
		tmp = ((t_0 * -2.0) * J_m) * ((0.5 / (math.fabs(t_0) * J_m)) * U_m)
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K * 0.5))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	t_4 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(t_4 * sqrt(Float64(0.25 * Float64((U_m ^ 2.0) / (t_4 ^ 2.0))))));
	elseif (t_3 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(t_0 * -2.0) * J_m) * Float64(Float64(0.5 / Float64(abs(t_0) * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((-0.5 * K));
	t_1 = cos((K * 0.5));
	t_2 = cos((K / 2.0));
	t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_2)) ^ 2.0)));
	t_4 = cos((0.5 * K));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -2.0 * (t_4 * sqrt((0.25 * ((U_m ^ 2.0) / (t_4 ^ 2.0)))));
	elseif (t_3 <= 1e+290)
		tmp = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	else
		tmp = ((t_0 * -2.0) * J_m) * ((0.5 / (abs(t_0) * J_m)) * U_m);
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(-2.0 * N[(t$95$4 * N[Sqrt[N[(0.25 * N[(N[Power[U$95$m, 2.0], $MachinePrecision] / N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+290], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[(N[(0.5 / N[(N[Abs[t$95$0], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $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 73.4%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 28.5

          \[\leadsto -2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 28.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.4

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

   3. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \cos \left(-0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(t\_3 \cdot \sqrt{0.25 \cdot \frac{{U\_m}^{2}}{0.5 + 0.5 \cdot \cos K}}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_3 \cdot -2\right) \cdot J\_m\right) \cdot \left(\frac{0.5}{\left|t\_3\right| \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0)))))
        (t_3 (cos (* -0.5 K))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (*
       -2.0
       (* t_3 (sqrt (* 0.25 (/ (pow U_m 2.0) (+ 0.5 (* 0.5 (cos K))))))))
      (if (<= t_2 1e+290)
        (*
         (* (* -2.0 J_m) t_0)
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
        (* (* (* t_3 -2.0) J_m) (* (/ 0.5 (* (fabs t_3) J_m)) U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double t_3 = cos((-0.5 * K));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (t_3 * sqrt((0.25 * (pow(U_m, 2.0) / (0.5 + (0.5 * cos(K)))))));
	} else if (t_2 <= 1e+290) {
		tmp = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	} else {
		tmp = ((t_3 * -2.0) * J_m) * ((0.5 / (fabs(t_3) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double t_3 = Math.cos((-0.5 * K));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (t_3 * Math.sqrt((0.25 * (Math.pow(U_m, 2.0) / (0.5 + (0.5 * Math.cos(K)))))));
	} else if (t_2 <= 1e+290) {
		tmp = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	} else {
		tmp = ((t_3 * -2.0) * J_m) * ((0.5 / (Math.abs(t_3) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K * 0.5))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	t_3 = math.cos((-0.5 * K))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (t_3 * math.sqrt((0.25 * (math.pow(U_m, 2.0) / (0.5 + (0.5 * math.cos(K)))))))
	elif t_2 <= 1e+290:
		tmp = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	else:
		tmp = ((t_3 * -2.0) * J_m) * ((0.5 / (math.fabs(t_3) * J_m)) * U_m)
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	t_3 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(t_3 * sqrt(Float64(0.25 * Float64((U_m ^ 2.0) / Float64(0.5 + Float64(0.5 * cos(K))))))));
	elseif (t_2 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(t_3 * -2.0) * J_m) * Float64(Float64(0.5 / Float64(abs(t_3) * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K * 0.5));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	t_3 = cos((-0.5 * K));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (t_3 * sqrt((0.25 * ((U_m ^ 2.0) / (0.5 + (0.5 * cos(K)))))));
	elseif (t_2 <= 1e+290)
		tmp = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	else
		tmp = ((t_3 * -2.0) * J_m) * ((0.5 / (abs(t_3) * J_m)) * U_m);
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(t$95$3 * N[Sqrt[N[(0.25 * N[(N[Power[U$95$m, 2.0], $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+290], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$3 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[(N[(0.5 / N[(N[Abs[t$95$3], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $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 73.4%

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

   3. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   4. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 28.5

          \[\leadsto -2 \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right) \]

   6. Applied rewrites 28.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.4

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

   3. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(t\_2 \cdot \sqrt{0.25 \cdot \frac{{U\_m}^{2}}{0.5 + 0.5 \cdot \cos K}}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + \frac{\frac{\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}}{4}}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 \cdot -2\right) \cdot J\_m\right) \cdot \left(\frac{0.5}{\left|t\_2\right| \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (cos (* -0.5 K))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (*
       -2.0
       (* t_2 (sqrt (* 0.25 (/ (pow U_m 2.0) (+ 0.5 (* 0.5 (cos K))))))))
      (if (<= t_1 1e+290)
        (*
         (* (* -2.0 J_m) (cos (* K 0.5)))
         (sqrt
          (+
           1.0
           (/ (/ (* (/ U_m J_m) (/ U_m J_m)) 4.0) (fma (cos K) 0.5 0.5)))))
        (* (* (* t_2 -2.0) J_m) (* (/ 0.5 (* (fabs t_2) J_m)) U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = cos((-0.5 * K));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -2.0 * (t_2 * sqrt((0.25 * (pow(U_m, 2.0) / (0.5 + (0.5 * cos(K)))))));
	} else if (t_1 <= 1e+290) {
		tmp = ((-2.0 * J_m) * cos((K * 0.5))) * sqrt((1.0 + ((((U_m / J_m) * (U_m / J_m)) / 4.0) / fma(cos(K), 0.5, 0.5))));
	} else {
		tmp = ((t_2 * -2.0) * J_m) * ((0.5 / (fabs(t_2) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(t_2 * sqrt(Float64(0.25 * Float64((U_m ^ 2.0) / Float64(0.5 + Float64(0.5 * cos(K))))))));
	elseif (t_1 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) / 4.0) / fma(cos(K), 0.5, 0.5)))));
	else
		tmp = Float64(Float64(Float64(t_2 * -2.0) * J_m) * Float64(Float64(0.5 / Float64(abs(t_2) * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(t$95$2 * N[Sqrt[N[(0.25 * N[(N[Power[U$95$m, 2.0], $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+290], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] / N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[(N[(0.5 / N[(N[Abs[t$95$2], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $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 73.4%

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

   3. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   4. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 28.5

          \[\leadsto -2 \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right) \]

   6. Applied rewrites 28.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.4

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

   3. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. frac-times N/A

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

      16. unpow2 N/A

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

      17. lift-pow.f64 N/A

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

   7. Applied rewrites 73.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}} \]

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\frac{\frac{\sqrt{0.25}}{\left|t\_0\right|} \cdot U\_m}{\left|J\_m\right|} \cdot \left(-2 \cdot J\_m\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_3 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + \frac{\frac{\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}}{4}}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot -2\right) \cdot J\_m\right) \cdot \left(\frac{0.5}{\left|t\_1\right| \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (cos (* -0.5 K)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0))))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (*
       (* (/ (* (/ (sqrt 0.25) (fabs t_0)) U_m) (fabs J_m)) (* -2.0 J_m))
       t_0)
      (if (<= t_3 1e+290)
        (*
         (* (* -2.0 J_m) (cos (* K 0.5)))
         (sqrt
          (+
           1.0
           (/ (/ (* (/ U_m J_m) (/ U_m J_m)) 4.0) (fma (cos K) 0.5 0.5)))))
        (* (* (* t_1 -2.0) J_m) (* (/ 0.5 (* (fabs t_1) J_m)) U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((-0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((((sqrt(0.25) / fabs(t_0)) * U_m) / fabs(J_m)) * (-2.0 * J_m)) * t_0;
	} else if (t_3 <= 1e+290) {
		tmp = ((-2.0 * J_m) * cos((K * 0.5))) * sqrt((1.0 + ((((U_m / J_m) * (U_m / J_m)) / 4.0) / fma(cos(K), 0.5, 0.5))));
	} else {
		tmp = ((t_1 * -2.0) * J_m) * ((0.5 / (fabs(t_1) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(0.25) / abs(t_0)) * U_m) / abs(J_m)) * Float64(-2.0 * J_m)) * t_0);
	elseif (t_3 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) / 4.0) / fma(cos(K), 0.5, 0.5)))));
	else
		tmp = Float64(Float64(Float64(t_1 * -2.0) * J_m) * Float64(Float64(0.5 / Float64(abs(t_1) * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(N[(N[(N[Sqrt[0.25], $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] / N[Abs[J$95$m], $MachinePrecision]), $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 1e+290], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] / N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[(N[(0.5 / N[(N[Abs[t$95$1], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   6. Applied rewrites 39.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\color{blue}{\left|J\right|}}\right) \]

   8. Applied rewrites 39.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.4

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

   3. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. frac-times N/A

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

      16. unpow2 N/A

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

      17. lift-pow.f64 N/A

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

   7. Applied rewrites 73.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}} \]

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\frac{\frac{\sqrt{0.25}}{\left|t\_0\right|} \cdot U\_m}{\left|J\_m\right|} \cdot \left(-2 \cdot J\_m\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_3 \leq 10^{+290}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m}, \frac{\frac{U\_m}{J\_m}}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot 4}, 1\right)} \cdot t\_1\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot -2\right) \cdot J\_m\right) \cdot \left(\frac{0.5}{\left|t\_1\right| \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (cos (* -0.5 K)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0))))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (*
       (* (/ (* (/ (sqrt 0.25) (fabs t_0)) U_m) (fabs J_m)) (* -2.0 J_m))
       t_0)
      (if (<= t_3 1e+290)
        (*
         (*
          (sqrt
           (fma (/ U_m J_m) (/ (/ U_m J_m) (* (fma (cos K) 0.5 0.5) 4.0)) 1.0))
          t_1)
         (* J_m -2.0))
        (* (* (* t_1 -2.0) J_m) (* (/ 0.5 (* (fabs t_1) J_m)) U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((-0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((((sqrt(0.25) / fabs(t_0)) * U_m) / fabs(J_m)) * (-2.0 * J_m)) * t_0;
	} else if (t_3 <= 1e+290) {
		tmp = (sqrt(fma((U_m / J_m), ((U_m / J_m) / (fma(cos(K), 0.5, 0.5) * 4.0)), 1.0)) * t_1) * (J_m * -2.0);
	} else {
		tmp = ((t_1 * -2.0) * J_m) * ((0.5 / (fabs(t_1) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(0.25) / abs(t_0)) * U_m) / abs(J_m)) * Float64(-2.0 * J_m)) * t_0);
	elseif (t_3 <= 1e+290)
		tmp = Float64(Float64(sqrt(fma(Float64(U_m / J_m), Float64(Float64(U_m / J_m) / Float64(fma(cos(K), 0.5, 0.5) * 4.0)), 1.0)) * t_1) * Float64(J_m * -2.0));
	else
		tmp = Float64(Float64(Float64(t_1 * -2.0) * J_m) * Float64(Float64(0.5 / Float64(abs(t_1) * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(N[(N[(N[Sqrt[0.25], $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] / N[Abs[J$95$m], $MachinePrecision]), $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 1e+290], N[(N[(N[Sqrt[N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(N[(U$95$m / J$95$m), $MachinePrecision] / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[(N[(0.5 / N[(N[Abs[t$95$1], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   6. Applied rewrites 39.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\color{blue}{\left|J\right|}}\right) \]

   8. Applied rewrites 39.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

   3. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 73.3%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, \frac{\frac{U}{J}}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot 4}, 1\right)}} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right) \]

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \left(t\_1 \cdot -2\right) \cdot J\_m\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(\left(-2 \cdot J\_m\right) \cdot t\_3\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_3}\right)}^{2}}\\ t_5 := \frac{U\_m}{J\_m + J\_m}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(\frac{\frac{\sqrt{0.25}}{\left|t\_0\right|} \cdot U\_m}{\left|J\_m\right|} \cdot \left(-2 \cdot J\_m\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(-2 \cdot t\_0\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(t\_5, t\_5, 1\right)}\\ \mathbf{elif}\;t\_4 \leq 10^{+290}:\\ \;\;\;\;t\_2 \cdot \sqrt{\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J\_m \cdot J\_m}, \frac{-0.25}{\mathsf{fma}\left(\cos K, -0.5, -0.5\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\frac{0.5}{\left|t\_1\right| \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (cos (* -0.5 K)))
        (t_2 (* (* t_1 -2.0) J_m))
        (t_3 (cos (/ K 2.0)))
        (t_4
         (*
          (* (* -2.0 J_m) t_3)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_3)) 2.0)))))
        (t_5 (/ U_m (+ J_m J_m))))
   (*
    J_s
    (if (<= t_4 (- INFINITY))
      (*
       (* (/ (* (/ (sqrt 0.25) (fabs t_0)) U_m) (fabs J_m)) (* -2.0 J_m))
       t_0)
      (if (<= t_4 2e-94)
        (* (* (* -2.0 t_0) J_m) (sqrt (fma t_5 t_5 1.0)))
        (if (<= t_4 1e+290)
          (*
           t_2
           (sqrt
            (fma
             (* U_m (/ U_m (* J_m J_m)))
             (/ -0.25 (fma (cos K) -0.5 -0.5))
             1.0)))
          (* t_2 (* (/ 0.5 (* (fabs t_1) J_m)) U_m))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((-0.5 * K));
	double t_2 = (t_1 * -2.0) * J_m;
	double t_3 = cos((K / 2.0));
	double t_4 = ((-2.0 * J_m) * t_3) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_3)), 2.0)));
	double t_5 = U_m / (J_m + J_m);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((((sqrt(0.25) / fabs(t_0)) * U_m) / fabs(J_m)) * (-2.0 * J_m)) * t_0;
	} else if (t_4 <= 2e-94) {
		tmp = ((-2.0 * t_0) * J_m) * sqrt(fma(t_5, t_5, 1.0));
	} else if (t_4 <= 1e+290) {
		tmp = t_2 * sqrt(fma((U_m * (U_m / (J_m * J_m))), (-0.25 / fma(cos(K), -0.5, -0.5)), 1.0));
	} else {
		tmp = t_2 * ((0.5 / (fabs(t_1) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = Float64(Float64(t_1 * -2.0) * J_m)
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(Float64(-2.0 * J_m) * t_3) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_3)) ^ 2.0))))
	t_5 = Float64(U_m / Float64(J_m + J_m))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(0.25) / abs(t_0)) * U_m) / abs(J_m)) * Float64(-2.0 * J_m)) * t_0);
	elseif (t_4 <= 2e-94)
		tmp = Float64(Float64(Float64(-2.0 * t_0) * J_m) * sqrt(fma(t_5, t_5, 1.0)));
	elseif (t_4 <= 1e+290)
		tmp = Float64(t_2 * sqrt(fma(Float64(U_m * Float64(U_m / Float64(J_m * J_m))), Float64(-0.25 / fma(cos(K), -0.5, -0.5)), 1.0)));
	else
		tmp = Float64(t_2 * Float64(Float64(0.5 / Float64(abs(t_1) * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(U$95$m / N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(N[(N[(N[Sqrt[0.25], $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] / N[Abs[J$95$m], $MachinePrecision]), $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$4, 2e-94], N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$5 * t$95$5 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+290], N[(t$95$2 * N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 / N[(N[Cos[K], $MachinePrecision] * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(0.5 / N[(N[Abs[t$95$1], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]]), $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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   6. Applied rewrites 39.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\color{blue}{\left|J\right|}}\right) \]

   8. Applied rewrites 39.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.9999999999999999e-94

   1. Initial program 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   8. Applied rewrites 64.7%

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

   if 1.9999999999999999e-94 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

   3. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   5. Applied rewrites 61.9%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \frac{U\_m}{J\_m + J\_m}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\frac{\frac{\sqrt{0.25}}{\left|t\_1\right|} \cdot U\_m}{\left|J\_m\right|} \cdot \left(-2 \cdot J\_m\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(\left(-2 \cdot t\_1\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(t\_4, t\_4, 1\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+290}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J\_m \cdot J\_m}, \frac{-0.25}{\mathsf{fma}\left(\cos K, -0.5, -0.5\right)}, 1\right)} \cdot t\_0\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot -2\right) \cdot J\_m\right) \cdot \left(\frac{0.5}{\left|t\_0\right| \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (* 0.5 K)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0)))))
        (t_4 (/ U_m (+ J_m J_m))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (*
       (* (/ (* (/ (sqrt 0.25) (fabs t_1)) U_m) (fabs J_m)) (* -2.0 J_m))
       t_1)
      (if (<= t_3 2e-94)
        (* (* (* -2.0 t_1) J_m) (sqrt (fma t_4 t_4 1.0)))
        (if (<= t_3 1e+290)
          (*
           (*
            (sqrt
             (fma
              (* U_m (/ U_m (* J_m J_m)))
              (/ -0.25 (fma (cos K) -0.5 -0.5))
              1.0))
            t_0)
           (* J_m -2.0))
          (* (* (* t_0 -2.0) J_m) (* (/ 0.5 (* (fabs t_0) J_m)) U_m))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double t_4 = U_m / (J_m + J_m);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((((sqrt(0.25) / fabs(t_1)) * U_m) / fabs(J_m)) * (-2.0 * J_m)) * t_1;
	} else if (t_3 <= 2e-94) {
		tmp = ((-2.0 * t_1) * J_m) * sqrt(fma(t_4, t_4, 1.0));
	} else if (t_3 <= 1e+290) {
		tmp = (sqrt(fma((U_m * (U_m / (J_m * J_m))), (-0.25 / fma(cos(K), -0.5, -0.5)), 1.0)) * t_0) * (J_m * -2.0);
	} else {
		tmp = ((t_0 * -2.0) * J_m) * ((0.5 / (fabs(t_0) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	t_4 = Float64(U_m / Float64(J_m + J_m))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(0.25) / abs(t_1)) * U_m) / abs(J_m)) * Float64(-2.0 * J_m)) * t_1);
	elseif (t_3 <= 2e-94)
		tmp = Float64(Float64(Float64(-2.0 * t_1) * J_m) * sqrt(fma(t_4, t_4, 1.0)));
	elseif (t_3 <= 1e+290)
		tmp = Float64(Float64(sqrt(fma(Float64(U_m * Float64(U_m / Float64(J_m * J_m))), Float64(-0.25 / fma(cos(K), -0.5, -0.5)), 1.0)) * t_0) * Float64(J_m * -2.0));
	else
		tmp = Float64(Float64(Float64(t_0 * -2.0) * J_m) * Float64(Float64(0.5 / Float64(abs(t_0) * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(U$95$m / N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(N[(N[(N[Sqrt[0.25], $MachinePrecision] / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] / N[Abs[J$95$m], $MachinePrecision]), $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 2e-94], N[(N[(N[(-2.0 * t$95$1), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$4 * t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+290], N[(N[(N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 / N[(N[Cos[K], $MachinePrecision] * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[(N[(0.5 / N[(N[Abs[t$95$0], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]]), $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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   6. Applied rewrites 39.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\color{blue}{\left|J\right|}}\right) \]

   8. Applied rewrites 39.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.9999999999999999e-94

   1. Initial program 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   8. Applied rewrites 64.7%

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

   if 1.9999999999999999e-94 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

   3. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 61.9%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \frac{U\_m}{J\_m + J\_m}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\frac{\frac{\sqrt{0.25}}{\left|t\_0\right|} \cdot U\_m}{\left|J\_m\right|} \cdot \left(-2 \cdot J\_m\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_3 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot t\_0\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(t\_4, t\_4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot -2\right) \cdot J\_m\right) \cdot \left(\frac{0.5}{\left|t\_1\right| \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (cos (* -0.5 K)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0)))))
        (t_4 (/ U_m (+ J_m J_m))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (*
       (* (/ (* (/ (sqrt 0.25) (fabs t_0)) U_m) (fabs J_m)) (* -2.0 J_m))
       t_0)
      (if (<= t_3 1e+290)
        (* (* (* -2.0 t_0) J_m) (sqrt (fma t_4 t_4 1.0)))
        (* (* (* t_1 -2.0) J_m) (* (/ 0.5 (* (fabs t_1) J_m)) U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((-0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double t_4 = U_m / (J_m + J_m);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((((sqrt(0.25) / fabs(t_0)) * U_m) / fabs(J_m)) * (-2.0 * J_m)) * t_0;
	} else if (t_3 <= 1e+290) {
		tmp = ((-2.0 * t_0) * J_m) * sqrt(fma(t_4, t_4, 1.0));
	} else {
		tmp = ((t_1 * -2.0) * J_m) * ((0.5 / (fabs(t_1) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	t_4 = Float64(U_m / Float64(J_m + J_m))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(0.25) / abs(t_0)) * U_m) / abs(J_m)) * Float64(-2.0 * J_m)) * t_0);
	elseif (t_3 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * t_0) * J_m) * sqrt(fma(t_4, t_4, 1.0)));
	else
		tmp = Float64(Float64(Float64(t_1 * -2.0) * J_m) * Float64(Float64(0.5 / Float64(abs(t_1) * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(U$95$m / N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(N[(N[(N[Sqrt[0.25], $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] / N[Abs[J$95$m], $MachinePrecision]), $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 1e+290], N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$4 * t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[(N[(0.5 / N[(N[Abs[t$95$1], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   6. Applied rewrites 39.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\color{blue}{\left|J\right|}}\right) \]

   8. Applied rewrites 39.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   8. Applied rewrites 64.7%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \frac{U\_m}{J\_m + J\_m}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\frac{\frac{\sqrt{0.25}}{\left|t\_0\right|} \cdot U\_m}{\left|J\_m\right|} \cdot t\_0\right) \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot t\_0\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(t\_4, t\_4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot -2\right) \cdot J\_m\right) \cdot \left(\frac{0.5}{\left|t\_1\right| \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (cos (* -0.5 K)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0)))))
        (t_4 (/ U_m (+ J_m J_m))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (*
       (* (/ (* (/ (sqrt 0.25) (fabs t_0)) U_m) (fabs J_m)) t_0)
       (* -2.0 J_m))
      (if (<= t_3 1e+290)
        (* (* (* -2.0 t_0) J_m) (sqrt (fma t_4 t_4 1.0)))
        (* (* (* t_1 -2.0) J_m) (* (/ 0.5 (* (fabs t_1) J_m)) U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((-0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double t_4 = U_m / (J_m + J_m);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((((sqrt(0.25) / fabs(t_0)) * U_m) / fabs(J_m)) * t_0) * (-2.0 * J_m);
	} else if (t_3 <= 1e+290) {
		tmp = ((-2.0 * t_0) * J_m) * sqrt(fma(t_4, t_4, 1.0));
	} else {
		tmp = ((t_1 * -2.0) * J_m) * ((0.5 / (fabs(t_1) * J_m)) * U_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	t_4 = Float64(U_m / Float64(J_m + J_m))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(0.25) / abs(t_0)) * U_m) / abs(J_m)) * t_0) * Float64(-2.0 * J_m));
	elseif (t_3 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * t_0) * J_m) * sqrt(fma(t_4, t_4, 1.0)));
	else
		tmp = Float64(Float64(Float64(t_1 * -2.0) * J_m) * Float64(Float64(0.5 / Float64(abs(t_1) * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(U$95$m / N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(N[(N[(N[Sqrt[0.25], $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] / N[Abs[J$95$m], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+290], N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$4 * t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[(N[(0.5 / N[(N[Abs[t$95$1], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   6. Applied rewrites 39.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\color{blue}{\left|J\right|}}\right) \]

   8. Applied rewrites 39.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   8. Applied rewrites 64.7%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\_m\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ t_4 := \frac{U\_m}{J\_m + J\_m}\\ t_5 := \left|t\_0\right|\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2 \cdot \left(\frac{0.5}{t\_5} \cdot \frac{U\_m}{J\_m}\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(t\_4, t\_4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot -2\right) \cdot J\_m\right) \cdot \left(\frac{0.5}{t\_5 \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J_m) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0)))))
        (t_4 (/ U_m (+ J_m J_m)))
        (t_5 (fabs t_0)))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (* t_2 (* (/ 0.5 t_5) (/ U_m J_m)))
      (if (<= t_3 1e+290)
        (* (* (* -2.0 (cos (* 0.5 K))) J_m) (sqrt (fma t_4 t_4 1.0)))
        (* (* (* t_0 -2.0) J_m) (* (/ 0.5 (* t_5 J_m)) U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J_m) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double t_4 = U_m / (J_m + J_m);
	double t_5 = fabs(t_0);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2 * ((0.5 / t_5) * (U_m / J_m));
	} else if (t_3 <= 1e+290) {
		tmp = ((-2.0 * cos((0.5 * K))) * J_m) * sqrt(fma(t_4, t_4, 1.0));
	} else {
		tmp = ((t_0 * -2.0) * J_m) * ((0.5 / (t_5 * J_m)) * U_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J_m) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	t_4 = Float64(U_m / Float64(J_m + J_m))
	t_5 = abs(t_0)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(t_2 * Float64(Float64(0.5 / t_5) * Float64(U_m / J_m)));
	elseif (t_3 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * cos(Float64(0.5 * K))) * J_m) * sqrt(fma(t_4, t_4, 1.0)));
	else
		tmp = Float64(Float64(Float64(t_0 * -2.0) * J_m) * Float64(Float64(0.5 / Float64(t_5 * J_m)) * U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(U$95$m / N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Abs[t$95$0], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(t$95$2 * N[(N[(0.5 / t$95$5), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+290], N[(N[(N[(-2.0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$4 * t$95$4 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[(N[(0.5 / N[(t$95$5 * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 39.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   8. Applied rewrites 64.7%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \frac{U\_m}{J\_m + J\_m}\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \frac{0.5}{\left|t\_1\right| \cdot J\_m} \cdot U\_m\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(t\_4 \cdot \left(J\_m \cdot -2\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 \cdot -2\right) \cdot J\_m\right) \cdot t\_4\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (/ U_m (+ J_m J_m)))
        (t_1 (cos (* -0.5 K)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0)))))
        (t_4 (* (/ 0.5 (* (fabs t_1) J_m)) U_m)))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (* (* t_4 (* J_m -2.0)) t_1)
      (if (<= t_3 1e+290)
        (* (* (* -2.0 (cos (* 0.5 K))) J_m) (sqrt (fma t_0 t_0 1.0)))
        (* (* (* t_1 -2.0) J_m) t_4))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = U_m / (J_m + J_m);
	double t_1 = cos((-0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double t_4 = (0.5 / (fabs(t_1) * J_m)) * U_m;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (t_4 * (J_m * -2.0)) * t_1;
	} else if (t_3 <= 1e+290) {
		tmp = ((-2.0 * cos((0.5 * K))) * J_m) * sqrt(fma(t_0, t_0, 1.0));
	} else {
		tmp = ((t_1 * -2.0) * J_m) * t_4;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(U_m / Float64(J_m + J_m))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	t_4 = Float64(Float64(0.5 / Float64(abs(t_1) * J_m)) * U_m)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(t_4 * Float64(J_m * -2.0)) * t_1);
	elseif (t_3 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * cos(Float64(0.5 * K))) * J_m) * sqrt(fma(t_0, t_0, 1.0)));
	else
		tmp = Float64(Float64(Float64(t_1 * -2.0) * J_m) * t_4);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(U$95$m / N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.5 / N[(N[Abs[t$95$1], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(N[(t$95$4 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 1e+290], N[(N[(N[(-2.0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * t$95$4), $MachinePrecision]]]), $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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   8. Applied rewrites 64.7%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \frac{U\_m}{J\_m + J\_m}\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \frac{0.5}{\left|t\_1\right| \cdot J\_m} \cdot U\_m\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(t\_4 \cdot \left(J\_m \cdot -2\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 \cdot t\_1\right) \cdot \left(J\_m \cdot -2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (/ U_m (+ J_m J_m)))
        (t_1 (cos (* -0.5 K)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0)))))
        (t_4 (* (/ 0.5 (* (fabs t_1) J_m)) U_m)))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (* (* t_4 (* J_m -2.0)) t_1)
      (if (<= t_3 1e+290)
        (* (* (* -2.0 (cos (* 0.5 K))) J_m) (sqrt (fma t_0 t_0 1.0)))
        (* (* t_4 t_1) (* J_m -2.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = U_m / (J_m + J_m);
	double t_1 = cos((-0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double t_4 = (0.5 / (fabs(t_1) * J_m)) * U_m;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (t_4 * (J_m * -2.0)) * t_1;
	} else if (t_3 <= 1e+290) {
		tmp = ((-2.0 * cos((0.5 * K))) * J_m) * sqrt(fma(t_0, t_0, 1.0));
	} else {
		tmp = (t_4 * t_1) * (J_m * -2.0);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(U_m / Float64(J_m + J_m))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	t_4 = Float64(Float64(0.5 / Float64(abs(t_1) * J_m)) * U_m)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(t_4 * Float64(J_m * -2.0)) * t_1);
	elseif (t_3 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * cos(Float64(0.5 * K))) * J_m) * sqrt(fma(t_0, t_0, 1.0)));
	else
		tmp = Float64(Float64(t_4 * t_1) * Float64(J_m * -2.0));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(U$95$m / N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.5 / N[(N[Abs[t$95$1], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(N[(t$95$4 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 1e+290], N[(N[(N[(-2.0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 * t$95$1), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]]]), $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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   8. Applied rewrites 64.7%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.4%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{\left|\cos \left(-0.5 \cdot K\right)\right| \cdot J} \cdot U\right) \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \frac{U\_m}{J\_m + J\_m}\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \left(\left(\frac{0.5}{\left|t\_1\right| \cdot J\_m} \cdot U\_m\right) \cdot t\_1\right) \cdot \left(J\_m \cdot -2\right)\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(\left(-2 \cdot J\_m\right) \cdot t\_3\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_3}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (/ U_m (+ J_m J_m)))
        (t_1 (cos (* -0.5 K)))
        (t_2 (* (* (* (/ 0.5 (* (fabs t_1) J_m)) U_m) t_1) (* J_m -2.0)))
        (t_3 (cos (/ K 2.0)))
        (t_4
         (*
          (* (* -2.0 J_m) t_3)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_3)) 2.0))))))
   (*
    J_s
    (if (<= t_4 (- INFINITY))
      t_2
      (if (<= t_4 1e+290)
        (* (* (* -2.0 (cos (* 0.5 K))) J_m) (sqrt (fma t_0 t_0 1.0)))
        t_2)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = U_m / (J_m + J_m);
	double t_1 = cos((-0.5 * K));
	double t_2 = (((0.5 / (fabs(t_1) * J_m)) * U_m) * t_1) * (J_m * -2.0);
	double t_3 = cos((K / 2.0));
	double t_4 = ((-2.0 * J_m) * t_3) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_3)), 2.0)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_4 <= 1e+290) {
		tmp = ((-2.0 * cos((0.5 * K))) * J_m) * sqrt(fma(t_0, t_0, 1.0));
	} else {
		tmp = t_2;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(U_m / Float64(J_m + J_m))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = Float64(Float64(Float64(Float64(0.5 / Float64(abs(t_1) * J_m)) * U_m) * t_1) * Float64(J_m * -2.0))
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(Float64(-2.0 * J_m) * t_3) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_3)) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_4 <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * cos(Float64(0.5 * K))) * J_m) * sqrt(fma(t_0, t_0, 1.0)));
	else
		tmp = t_2;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(U$95$m / N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(0.5 / N[(N[Abs[t$95$1], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$4, (-Infinity)], t$95$2, If[LessEqual[t$95$4, 1e+290], N[(N[(N[(-2.0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 1.00000000000000006e290 < (*.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 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

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

   9. Applied rewrites 39.4%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

   8. Applied rewrites 64.7%

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

Alternative 14: 72.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := 1 + -0.125 \cdot {K}^{2}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}} \leq 10^{+290}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot J\_m\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U\_m}{\left(J\_m + J\_m\right) \cdot t\_0}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.125 (pow K 2.0)))) (t_1 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))
         1e+290)
      (* (* (* -2.0 J_m) (cos (* K 0.5))) (cosh (asinh (* 0.5 (/ U_m J_m)))))
      (* (* (* t_0 J_m) -2.0) (cosh (asinh (/ U_m (* (+ J_m J_m) t_0)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = 1.0 + (-0.125 * pow(K, 2.0));
	double t_1 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))) <= 1e+290) {
		tmp = ((-2.0 * J_m) * cos((K * 0.5))) * cosh(asinh((0.5 * (U_m / J_m))));
	} else {
		tmp = ((t_0 * J_m) * -2.0) * cosh(asinh((U_m / ((J_m + J_m) * t_0))));
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = 1.0 + (-0.125 * math.pow(K, 2.0))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))) <= 1e+290:
		tmp = ((-2.0 * J_m) * math.cos((K * 0.5))) * math.cosh(math.asinh((0.5 * (U_m / J_m))))
	else:
		tmp = ((t_0 * J_m) * -2.0) * math.cosh(math.asinh((U_m / ((J_m + J_m) * t_0))))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(1.0 + Float64(-0.125 * (K ^ 2.0)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0)))) <= 1e+290)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5))) * cosh(asinh(Float64(0.5 * Float64(U_m / J_m)))));
	else
		tmp = Float64(Float64(Float64(t_0 * J_m) * -2.0) * cosh(asinh(Float64(U_m / Float64(Float64(J_m + J_m) * t_0)))));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = 1.0 + (-0.125 * (K ^ 2.0));
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)))) <= 1e+290)
		tmp = ((-2.0 * J_m) * cos((K * 0.5))) * cosh(asinh((0.5 * (U_m / J_m))));
	else
		tmp = ((t_0 * J_m) * -2.0) * cosh(asinh((U_m / ((J_m + J_m) * t_0))));
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+290], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U$95$m / N[(N[(J$95$m + J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000006e290

   1. Initial program 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6

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

   6. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 71.6

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

   8. Applied rewrites 71.6%

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

   if 1.00000000000000006e290 < (*.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 73.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 84.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 84.7

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   3. Applied rewrites 84.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 84.7

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-neg-out N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in K around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\frac{-1}{8} \cdot {K}^{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot \color{blue}{{K}^{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \]

      3. lower-pow.f64 44.5

         \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]

   8. Applied rewrites 44.5%

      \[\leadsto \left(\left(\color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]

   9. Taylor expanded in K around 0

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

       1. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + \color{blue}{\frac{-1}{8} \cdot {K}^{2}}\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + \frac{-1}{8} \cdot \color{blue}{{K}^{2}}\right)}\right) \]

       3. lower-pow.f64 47.3

          \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right)}\right) \]

   11. Applied rewrites 47.3%

       \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 72.6% accurate, 0.6× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (/ U_m (+ J_m J_m))) (t_1 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))
         (- INFINITY))
      (* (* (cos (* -0.5 K)) (/ (* 0.5 U_m) (fabs J_m))) (* J_m -2.0))
      (* (* (* -2.0 (cos (* 0.5 K))) J_m) (sqrt (fma t_0 t_0 1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = U_m / (J_m + J_m);
	double t_1 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))) <= -((double) INFINITY)) {
		tmp = (cos((-0.5 * K)) * ((0.5 * U_m) / fabs(J_m))) * (J_m * -2.0);
	} else {
		tmp = ((-2.0 * cos((0.5 * K))) * J_m) * sqrt(fma(t_0, t_0, 1.0));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(U_m / Float64(J_m + J_m))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(Float64(cos(Float64(-0.5 * K)) * Float64(Float64(0.5 * U_m) / abs(J_m))) * Float64(J_m * -2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 * cos(Float64(0.5 * K))) * J_m) * sqrt(fma(t_0, t_0, 1.0)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(U$95$m / N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * U$95$m), $MachinePrecision] / N[Abs[J$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.4%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right)} \]

   6. Applied rewrites 39.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\color{blue}{\left|J\right|}}\right) \]

   7. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|\color{blue}{J}\right|}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 25.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{0.5}{\left|\color{blue}{J}\right|}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right)\right) \cdot \left(-2 \cdot J\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right)\right) \cdot \left(-2 \cdot J\right)} \]

   11. Applied rewrites 25.2%

       \[\leadsto \color{blue}{\left(\cos \left(-0.5 \cdot K\right) \cdot \frac{0.5 \cdot U}{\left|J\right|}\right) \cdot \left(J \cdot -2\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} + 1} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}} + 1} \]

      6. cosh-asinh-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      7. lower-cosh.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      8. lower-asinh.f64 84.7

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \color{blue}{\sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      10. count-2-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      11. lower-+.f64 84.7

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right) \]

      13. cos-neg-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right)}\right) \]

      16. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{\color{blue}{-2}}\right)}\right) \]

      18. mult-flip-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{-2}\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      21. metadata-eval 84.7

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\color{blue}{-0.5} \cdot K\right)}\right) \]

   3. Applied rewrites 84.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)} \]

   4. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \color{blue}{\frac{U}{J}}\right) \]

      2. lower-/.f64 71.6

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\color{blue}{J}}\right) \]

   6. Applied rewrites 71.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(0.5 \cdot \frac{U}{J}\right)} \]

   8. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 71.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U\_m}{J\_m}\right)\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (* (* (* -2.0 J_m) (cos (* K 0.5))) (cosh (asinh (* 0.5 (/ U_m J_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (((-2.0 * J_m) * cos((K * 0.5))) * cosh(asinh((0.5 * (U_m / J_m)))));
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * (((-2.0 * J_m) * math.cos((K * 0.5))) * math.cosh(math.asinh((0.5 * (U_m / J_m)))))

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5))) * cosh(asinh(Float64(0.5 * Float64(U_m / J_m))))))
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * (((-2.0 * J_m) * cos((K * 0.5))) * cosh(asinh((0.5 * (U_m / J_m)))));
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.4%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

   3. +-commutative N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} + 1} \]

   5. unpow2 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}} + 1} \]

   6. cosh-asinh-rev N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

   7. lower-cosh.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

   8. lower-asinh.f64 84.7

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \color{blue}{\sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

   10. count-2-rev N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

   11. lower-+.f64 84.7

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

   12. lift-cos.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right) \]

   13. cos-neg-rev N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

   15. lift-/.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right)}\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{\color{blue}{-2}}\right)}\right) \]

   18. mult-flip-rev N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{-2}\right)}}\right) \]

   19. *-commutative N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

   21. metadata-eval 84.7

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\color{blue}{-0.5} \cdot K\right)}\right) \]

3. Applied rewrites 84.7%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)} \]

4. Taylor expanded in K around 0

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \color{blue}{\frac{U}{J}}\right) \]

   2. lower-/.f64 71.6

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\color{blue}{J}}\right) \]

6. Applied rewrites 71.6%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(0.5 \cdot \frac{U}{J}\right)} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right) \]

   2. mult-flip N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right) \]

   4. lower-*.f64 71.6

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{J}\right) \]

8. Applied rewrites 71.6%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{J}\right) \]
9. Add Preprocessing

Alternative 17: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ t_3 := t\_2 \cdot \left(J\_m \cdot -2\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_2 \cdot \frac{0.5 \cdot U\_m}{\left|J\_m\right|}\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+143}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-125}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \sqrt{1 + 0.25 \cdot \frac{{U\_m}^{2}}{{J\_m}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (cos (* -0.5 K)))
        (t_3 (* t_2 (* J_m -2.0))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (* (* t_2 (/ (* 0.5 U_m) (fabs J_m))) (* J_m -2.0))
      (if (<= t_1 -2e+143)
        t_3
        (if (<= t_1 -5e-125)
          (*
           -2.0
           (* J_m (sqrt (+ 1.0 (* 0.25 (/ (pow U_m 2.0) (pow J_m 2.0)))))))
          t_3))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = cos((-0.5 * K));
	double t_3 = t_2 * (J_m * -2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_2 * ((0.5 * U_m) / fabs(J_m))) * (J_m * -2.0);
	} else if (t_1 <= -2e+143) {
		tmp = t_3;
	} else if (t_1 <= -5e-125) {
		tmp = -2.0 * (J_m * sqrt((1.0 + (0.25 * (pow(U_m, 2.0) / pow(J_m, 2.0))))));
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = Math.cos((-0.5 * K));
	double t_3 = t_2 * (J_m * -2.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_2 * ((0.5 * U_m) / Math.abs(J_m))) * (J_m * -2.0);
	} else if (t_1 <= -2e+143) {
		tmp = t_3;
	} else if (t_1 <= -5e-125) {
		tmp = -2.0 * (J_m * Math.sqrt((1.0 + (0.25 * (Math.pow(U_m, 2.0) / Math.pow(J_m, 2.0))))));
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	t_2 = math.cos((-0.5 * K))
	t_3 = t_2 * (J_m * -2.0)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (t_2 * ((0.5 * U_m) / math.fabs(J_m))) * (J_m * -2.0)
	elif t_1 <= -2e+143:
		tmp = t_3
	elif t_1 <= -5e-125:
		tmp = -2.0 * (J_m * math.sqrt((1.0 + (0.25 * (math.pow(U_m, 2.0) / math.pow(J_m, 2.0))))))
	else:
		tmp = t_3
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(-0.5 * K))
	t_3 = Float64(t_2 * Float64(J_m * -2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_2 * Float64(Float64(0.5 * U_m) / abs(J_m))) * Float64(J_m * -2.0));
	elseif (t_1 <= -2e+143)
		tmp = t_3;
	elseif (t_1 <= -5e-125)
		tmp = Float64(-2.0 * Float64(J_m * sqrt(Float64(1.0 + Float64(0.25 * Float64((U_m ^ 2.0) / (J_m ^ 2.0)))))));
	else
		tmp = t_3;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	t_2 = cos((-0.5 * K));
	t_3 = t_2 * (J_m * -2.0);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (t_2 * ((0.5 * U_m) / abs(J_m))) * (J_m * -2.0);
	elseif (t_1 <= -2e+143)
		tmp = t_3;
	elseif (t_1 <= -5e-125)
		tmp = -2.0 * (J_m * sqrt((1.0 + (0.25 * ((U_m ^ 2.0) / (J_m ^ 2.0))))));
	else
		tmp = t_3;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(t$95$2 * N[(N[(0.5 * U$95$m), $MachinePrecision] / N[Abs[J$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+143], t$95$3, If[LessEqual[t$95$1, -5e-125], N[(-2.0 * N[(J$95$m * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[U$95$m, 2.0], $MachinePrecision] / N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]), $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 73.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around inf

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \color{blue}{\sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right)} \]

   6. Applied rewrites 39.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\color{blue}{\left|J\right|}}\right) \]

   7. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|\color{blue}{J}\right|}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 25.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{0.5}{\left|\color{blue}{J}\right|}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right)\right) \cdot \left(-2 \cdot J\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \left(U \cdot \frac{\frac{1}{2}}{\left|J\right|}\right)\right) \cdot \left(-2 \cdot J\right)} \]

   11. Applied rewrites 25.2%

       \[\leadsto \color{blue}{\left(\cos \left(-0.5 \cdot K\right) \cdot \frac{0.5 \cdot U}{\left|J\right|}\right) \cdot \left(J \cdot -2\right)} \]

   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))))) < -2e143 or -4.99999999999999967e-125 < (*.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 73.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   4. Taylor expanded in J around inf

      \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)} \cdot \left(J \cdot -2\right) \]
   5. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \cos \left(\frac{-1}{2} \cdot K\right) \cdot \left(J \cdot -2\right) \]

      2. lower-*.f64 52.4

         \[\leadsto \cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right) \]

   6. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\cos \left(-0.5 \cdot K\right)} \cdot \left(J \cdot -2\right) \]

   if -2e143 < (*.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))))) < -4.99999999999999967e-125

   1. Initial program 73.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      8. lower-pow.f64 33.9

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

   4. Applied rewrites 33.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right) \cdot \left(J\_m \cdot -2\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\_m\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2 \cdot \frac{0.5 \cdot U\_m}{J\_m}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-125}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \sqrt{1 + 0.25 \cdot \frac{{U\_m}^{2}}{{J\_m}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (cos (* -0.5 K)) (* J_m -2.0)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J_m) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (* t_2 (/ (* 0.5 U_m) J_m))
      (if (<= t_3 -2e+143)
        t_0
        (if (<= t_3 -5e-125)
          (*
           -2.0
           (* J_m (sqrt (+ 1.0 (* 0.25 (/ (pow U_m 2.0) (pow J_m 2.0)))))))
          t_0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((-0.5 * K)) * (J_m * -2.0);
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J_m) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2 * ((0.5 * U_m) / J_m);
	} else if (t_3 <= -2e+143) {
		tmp = t_0;
	} else if (t_3 <= -5e-125) {
		tmp = -2.0 * (J_m * sqrt((1.0 + (0.25 * (pow(U_m, 2.0) / pow(J_m, 2.0))))));
	} else {
		tmp = t_0;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((-0.5 * K)) * (J_m * -2.0);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = (-2.0 * J_m) * t_1;
	double t_3 = t_2 * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2 * ((0.5 * U_m) / J_m);
	} else if (t_3 <= -2e+143) {
		tmp = t_0;
	} else if (t_3 <= -5e-125) {
		tmp = -2.0 * (J_m * Math.sqrt((1.0 + (0.25 * (Math.pow(U_m, 2.0) / Math.pow(J_m, 2.0))))));
	} else {
		tmp = t_0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((-0.5 * K)) * (J_m * -2.0)
	t_1 = math.cos((K / 2.0))
	t_2 = (-2.0 * J_m) * t_1
	t_3 = t_2 * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_2 * ((0.5 * U_m) / J_m)
	elif t_3 <= -2e+143:
		tmp = t_0
	elif t_3 <= -5e-125:
		tmp = -2.0 * (J_m * math.sqrt((1.0 + (0.25 * (math.pow(U_m, 2.0) / math.pow(J_m, 2.0))))))
	else:
		tmp = t_0
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(cos(Float64(-0.5 * K)) * Float64(J_m * -2.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J_m) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(t_2 * Float64(Float64(0.5 * U_m) / J_m));
	elseif (t_3 <= -2e+143)
		tmp = t_0;
	elseif (t_3 <= -5e-125)
		tmp = Float64(-2.0 * Float64(J_m * sqrt(Float64(1.0 + Float64(0.25 * Float64((U_m ^ 2.0) / (J_m ^ 2.0)))))));
	else
		tmp = t_0;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((-0.5 * K)) * (J_m * -2.0);
	t_1 = cos((K / 2.0));
	t_2 = (-2.0 * J_m) * t_1;
	t_3 = t_2 * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_2 * ((0.5 * U_m) / J_m);
	elseif (t_3 <= -2e+143)
		tmp = t_0;
	elseif (t_3 <= -5e-125)
		tmp = -2.0 * (J_m * sqrt((1.0 + (0.25 * ((U_m ^ 2.0) / (J_m ^ 2.0))))));
	else
		tmp = t_0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], N[(t$95$2 * N[(N[(0.5 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e+143], t$95$0, If[LessEqual[t$95$3, -5e-125], N[(-2.0 * N[(J$95$m * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[U$95$m, 2.0], $MachinePrecision] / N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $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 73.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around inf

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \color{blue}{\sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      8. lower-*.f64 24.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 24.4%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right)} \]

   5. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-*.f64 39.4

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   7. Applied rewrites 39.4%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

   8. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\frac{1}{2} \cdot U}{J} \]
   9. Step-by-step derivation

      1. lower-*.f64 25.2

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{0.5 \cdot U}{J} \]

   10. Applied rewrites 25.2%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{0.5 \cdot U}{J} \]

   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))))) < -2e143 or -4.99999999999999967e-125 < (*.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 73.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   4. Taylor expanded in J around inf

      \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)} \cdot \left(J \cdot -2\right) \]
   5. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \cos \left(\frac{-1}{2} \cdot K\right) \cdot \left(J \cdot -2\right) \]

      2. lower-*.f64 52.4

         \[\leadsto \cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right) \]

   6. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\cos \left(-0.5 \cdot K\right)} \cdot \left(J \cdot -2\right) \]

   if -2e143 < (*.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))))) < -4.99999999999999967e-125

   1. Initial program 73.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      8. lower-pow.f64 33.9

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

   4. Applied rewrites 33.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 6.6 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \sqrt{1 + 0.25 \cdot \frac{{U\_m}^{2}}{{J\_m}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-0.5 \cdot K\right) \cdot \left(J\_m \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= K 6.6e-37)
    (* -2.0 (* J_m (sqrt (+ 1.0 (* 0.25 (/ (pow U_m 2.0) (pow J_m 2.0)))))))
    (* (cos (* -0.5 K)) (* J_m -2.0)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (K <= 6.6e-37) {
		tmp = -2.0 * (J_m * sqrt((1.0 + (0.25 * (pow(U_m, 2.0) / pow(J_m, 2.0))))));
	} else {
		tmp = cos((-0.5 * K)) * (J_m * -2.0);
	}
	return J_s * tmp;
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (k <= 6.6d-37) then
        tmp = (-2.0d0) * (j_m * sqrt((1.0d0 + (0.25d0 * ((u_m ** 2.0d0) / (j_m ** 2.0d0))))))
    else
        tmp = cos(((-0.5d0) * k)) * (j_m * (-2.0d0))
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (K <= 6.6e-37) {
		tmp = -2.0 * (J_m * Math.sqrt((1.0 + (0.25 * (Math.pow(U_m, 2.0) / Math.pow(J_m, 2.0))))));
	} else {
		tmp = Math.cos((-0.5 * K)) * (J_m * -2.0);
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if K <= 6.6e-37:
		tmp = -2.0 * (J_m * math.sqrt((1.0 + (0.25 * (math.pow(U_m, 2.0) / math.pow(J_m, 2.0))))))
	else:
		tmp = math.cos((-0.5 * K)) * (J_m * -2.0)
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (K <= 6.6e-37)
		tmp = Float64(-2.0 * Float64(J_m * sqrt(Float64(1.0 + Float64(0.25 * Float64((U_m ^ 2.0) / (J_m ^ 2.0)))))));
	else
		tmp = Float64(cos(Float64(-0.5 * K)) * Float64(J_m * -2.0));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (K <= 6.6e-37)
		tmp = -2.0 * (J_m * sqrt((1.0 + (0.25 * ((U_m ^ 2.0) / (J_m ^ 2.0))))));
	else
		tmp = cos((-0.5 * K)) * (J_m * -2.0);
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[K, 6.6e-37], N[(-2.0 * N[(J$95$m * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[U$95$m, 2.0], $MachinePrecision] / N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 6.59999999999999964e-37

   1. Initial program 73.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

      8. lower-pow.f64 33.9

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

   4. Applied rewrites 33.9%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

   if 6.59999999999999964e-37 < K

   1. Initial program 73.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   4. Taylor expanded in J around inf

      \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)} \cdot \left(J \cdot -2\right) \]
   5. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \cos \left(\frac{-1}{2} \cdot K\right) \cdot \left(J \cdot -2\right) \]

      2. lower-*.f64 52.4

         \[\leadsto \cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right) \]

   6. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\cos \left(-0.5 \cdot K\right)} \cdot \left(J \cdot -2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 44.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \left(J\_m \cdot -2\right)\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (* (cos (* -0.5 K)) (* J_m -2.0))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (cos((-0.5 * K)) * (J_m * -2.0));
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * (cos(((-0.5d0) * k)) * (j_m * (-2.0d0)))
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (Math.cos((-0.5 * K)) * (J_m * -2.0));
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * (math.cos((-0.5 * K)) * (J_m * -2.0))

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(cos(Float64(-0.5 * K)) * Float64(J_m * -2.0)))
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * (cos((-0.5 * K)) * (J_m * -2.0));
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.4%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

3. Applied rewrites 73.2%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

4. Taylor expanded in J around inf

   \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)} \cdot \left(J \cdot -2\right) \]
5. Step-by-step derivation

   1. lower-cos.f64 N/A

      \[\leadsto \cos \left(\frac{-1}{2} \cdot K\right) \cdot \left(J \cdot -2\right) \]

   2. lower-*.f64 52.4

      \[\leadsto \cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right) \]

6. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\cos \left(-0.5 \cdot K\right)} \cdot \left(J \cdot -2\right) \]
7. Add Preprocessing

Alternative 21: 28.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(\mathsf{fma}\left(-2, J\_m, \left(\left(0.25 \cdot J\_m\right) \cdot K\right) \cdot K\right) \cdot 1\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (* (fma -2.0 J_m (* (* (* 0.25 J_m) K) K)) 1.0)))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (fma(-2.0, J_m, (((0.25 * J_m) * K) * K)) * 1.0);
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(fma(-2.0, J_m, Float64(Float64(Float64(0.25 * J_m) * K) * K)) * 1.0))
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * N[(N[(-2.0 * J$95$m + N[(N[(N[(0.25 * J$95$m), $MachinePrecision] * K), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.4%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

2. Taylor expanded in J around inf

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 52.4%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]

5. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\left(-2 \cdot J + \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot 1 \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, \color{blue}{J}, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   4. lower-pow.f64 28.3

      \[\leadsto \mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

7. Applied rewrites 28.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot 1 \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\frac{1}{4} \cdot J\right) \cdot {K}^{2}\right) \cdot 1 \]

   4. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\frac{1}{4} \cdot J\right) \cdot {K}^{2}\right) \cdot 1 \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\frac{1}{4} \cdot J\right) \cdot \left(K \cdot K\right)\right) \cdot 1 \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(\frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(\frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(\frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]

   9. lower-*.f64 28.3

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]

9. Applied rewrites 28.3%

   \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(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)))))