Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.2% → 97.2%

Time: 7.5s

Alternatives: 15

Speedup: 0.3×

Specification

Math

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

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: 72.2% accurate, 1.0× speedup

Math

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

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: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := -2 \cdot \left(\left|U\right| \cdot \left(t\_1 \cdot \sqrt{\frac{0.25}{{t\_1}^{2}}}\right)\right)\\ t_3 := 2 \cdot \left|J\right|\\ t_4 := -2 \cdot \left|J\right|\\ t_5 := \left(t\_4 \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{t\_3 \cdot t\_0}\right)}^{2}}\\ t_6 := \cos \left(K \cdot 0.5\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 10^{+238}:\\ \;\;\;\;\left(t\_4 \cdot t\_6\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{t\_3 \cdot t\_6}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0)))
       (t_1 (cos (* 0.5 K)))
       (t_2
        (* -2.0 (* (fabs U) (* t_1 (sqrt (/ 0.25 (pow t_1 2.0)))))))
       (t_3 (* 2.0 (fabs J)))
       (t_4 (* -2.0 (fabs J)))
       (t_5
        (*
         (* t_4 t_0)
         (sqrt (+ 1.0 (pow (/ (fabs U) (* t_3 t_0)) 2.0)))))
       (t_6 (cos (* K 0.5))))
  (*
   (copysign 1.0 J)
   (if (<= t_5 (- INFINITY))
     t_2
     (if (<= t_5 1e+238)
       (*
        (* t_4 t_6)
        (sqrt (+ 1.0 (pow (/ (fabs U) (* t_3 t_6)) 2.0))))
       t_2)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = cos((0.5 * K));
	double t_2 = -2.0 * (fabs(U) * (t_1 * sqrt((0.25 / pow(t_1, 2.0)))));
	double t_3 = 2.0 * fabs(J);
	double t_4 = -2.0 * fabs(J);
	double t_5 = (t_4 * t_0) * sqrt((1.0 + pow((fabs(U) / (t_3 * t_0)), 2.0)));
	double t_6 = cos((K * 0.5));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_5 <= 1e+238) {
		tmp = (t_4 * t_6) * sqrt((1.0 + pow((fabs(U) / (t_3 * t_6)), 2.0)));
	} else {
		tmp = t_2;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.cos((0.5 * K));
	double t_2 = -2.0 * (Math.abs(U) * (t_1 * Math.sqrt((0.25 / Math.pow(t_1, 2.0)))));
	double t_3 = 2.0 * Math.abs(J);
	double t_4 = -2.0 * Math.abs(J);
	double t_5 = (t_4 * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / (t_3 * t_0)), 2.0)));
	double t_6 = Math.cos((K * 0.5));
	double tmp;
	if (t_5 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_5 <= 1e+238) {
		tmp = (t_4 * t_6) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / (t_3 * t_6)), 2.0)));
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.cos((0.5 * K))
	t_2 = -2.0 * (math.fabs(U) * (t_1 * math.sqrt((0.25 / math.pow(t_1, 2.0)))))
	t_3 = 2.0 * math.fabs(J)
	t_4 = -2.0 * math.fabs(J)
	t_5 = (t_4 * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / (t_3 * t_0)), 2.0)))
	t_6 = math.cos((K * 0.5))
	tmp = 0
	if t_5 <= -math.inf:
		tmp = t_2
	elif t_5 <= 1e+238:
		tmp = (t_4 * t_6) * math.sqrt((1.0 + math.pow((math.fabs(U) / (t_3 * t_6)), 2.0)))
	else:
		tmp = t_2
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = cos(Float64(0.5 * K))
	t_2 = Float64(-2.0 * Float64(abs(U) * Float64(t_1 * sqrt(Float64(0.25 / (t_1 ^ 2.0))))))
	t_3 = Float64(2.0 * abs(J))
	t_4 = Float64(-2.0 * abs(J))
	t_5 = Float64(Float64(t_4 * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(t_3 * t_0)) ^ 2.0))))
	t_6 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_5 <= 1e+238)
		tmp = Float64(Float64(t_4 * t_6) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(t_3 * t_6)) ^ 2.0))));
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = cos((0.5 * K));
	t_2 = -2.0 * (abs(U) * (t_1 * sqrt((0.25 / (t_1 ^ 2.0)))));
	t_3 = 2.0 * abs(J);
	t_4 = -2.0 * abs(J);
	t_5 = (t_4 * t_0) * sqrt((1.0 + ((abs(U) / (t_3 * t_0)) ^ 2.0)));
	t_6 = cos((K * 0.5));
	tmp = 0.0;
	if (t_5 <= -Inf)
		tmp = t_2;
	elseif (t_5 <= 1e+238)
		tmp = (t_4 * t_6) * sqrt((1.0 + ((abs(U) / (t_3 * t_6)) ^ 2.0)));
	else
		tmp = t_2;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(N[Abs[U], $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(0.25 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, (-Infinity)], t$95$2, If[LessEqual[t$95$5, 1e+238], N[(N[(t$95$4 * t$95$6), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(t$95$3 * t$95$6), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $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 1e238 < (*.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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.1%

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

   7. Applied rewrites 27.1%

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

   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))))) < 1e238

   1. Initial program 72.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 72.2%

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

   3. Applied rewrites 72.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 72.2%

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

   5. Applied rewrites 72.2%

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

Alternative 2: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(0.5 \cdot K\right)\\ t_3 := -2 \cdot \left(\left|U\right| \cdot \left(t\_2 \cdot \sqrt{\frac{0.25}{{t\_2}^{2}}}\right)\right)\\ t_4 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{+238}:\\ \;\;\;\;\left(\sqrt{\frac{t\_4 \cdot t\_4}{\cos \left(1 \cdot K\right) \cdot 0.5 - -0.5} - -1} \cdot \left(\left|J\right| \cdot -2\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0)))
       (t_1
        (*
         (* (* -2.0 (fabs J)) t_0)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0)))))
       (t_2 (cos (* 0.5 K)))
       (t_3
        (* -2.0 (* (fabs U) (* t_2 (sqrt (/ 0.25 (pow t_2 2.0)))))))
       (t_4 (/ (fabs U) (+ (fabs J) (fabs J)))))
  (*
   (copysign 1.0 J)
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 1e+238)
       (*
        (*
         (sqrt
          (- (/ (* t_4 t_4) (- (* (cos (* 1.0 K)) 0.5) -0.5)) -1.0))
         (* (fabs J) -2.0))
        (cos (* K 0.5)))
       t_3)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double t_2 = cos((0.5 * K));
	double t_3 = -2.0 * (fabs(U) * (t_2 * sqrt((0.25 / pow(t_2, 2.0)))));
	double t_4 = fabs(U) / (fabs(J) + fabs(J));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 1e+238) {
		tmp = (sqrt((((t_4 * t_4) / ((cos((1.0 * K)) * 0.5) - -0.5)) - -1.0)) * (fabs(J) * -2.0)) * cos((K * 0.5));
	} else {
		tmp = t_3;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * Math.abs(J)) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_0)), 2.0)));
	double t_2 = Math.cos((0.5 * K));
	double t_3 = -2.0 * (Math.abs(U) * (t_2 * Math.sqrt((0.25 / Math.pow(t_2, 2.0)))));
	double t_4 = Math.abs(U) / (Math.abs(J) + Math.abs(J));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_1 <= 1e+238) {
		tmp = (Math.sqrt((((t_4 * t_4) / ((Math.cos((1.0 * K)) * 0.5) - -0.5)) - -1.0)) * (Math.abs(J) * -2.0)) * Math.cos((K * 0.5));
	} else {
		tmp = t_3;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * math.fabs(J)) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_0)), 2.0)))
	t_2 = math.cos((0.5 * K))
	t_3 = -2.0 * (math.fabs(U) * (t_2 * math.sqrt((0.25 / math.pow(t_2, 2.0)))))
	t_4 = math.fabs(U) / (math.fabs(J) + math.fabs(J))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_3
	elif t_1 <= 1e+238:
		tmp = (math.sqrt((((t_4 * t_4) / ((math.cos((1.0 * K)) * 0.5) - -0.5)) - -1.0)) * (math.fabs(J) * -2.0)) * math.cos((K * 0.5))
	else:
		tmp = t_3
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * abs(J)) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	t_3 = Float64(-2.0 * Float64(abs(U) * Float64(t_2 * sqrt(Float64(0.25 / (t_2 ^ 2.0))))))
	t_4 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 1e+238)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(t_4 * t_4) / Float64(Float64(cos(Float64(1.0 * K)) * 0.5) - -0.5)) - -1.0)) * Float64(abs(J) * -2.0)) * cos(Float64(K * 0.5)));
	else
		tmp = t_3;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * abs(J)) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_0)) ^ 2.0)));
	t_2 = cos((0.5 * K));
	t_3 = -2.0 * (abs(U) * (t_2 * sqrt((0.25 / (t_2 ^ 2.0)))));
	t_4 = abs(U) / (abs(J) + abs(J));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_3;
	elseif (t_1 <= 1e+238)
		tmp = (sqrt((((t_4 * t_4) / ((cos((1.0 * K)) * 0.5) - -0.5)) - -1.0)) * (abs(J) * -2.0)) * cos((K * 0.5));
	else
		tmp = t_3;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $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[(-2.0 * N[(N[Abs[U], $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(0.25 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 1e+238], N[(N[(N[Sqrt[N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] / N[(N[(N[Cos[N[(1.0 * K), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]), $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 1e238 < (*.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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.1%

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

   7. Applied rewrites 27.1%

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

   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))))) < 1e238

   1. Initial program 72.2%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 72.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{\left(\sin \left(0.5 \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(0.5 \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right)}{\sin \left(0.5 \cdot K\right) \cdot 0 - 1 \cdot \cos \left(-0.5 \cdot K\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}\right)}^{2}} \]

      4. flip-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}{\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) - \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}}\right)}^{2}} \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}{\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) - \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}}\right)}^{2}} \]

   5. Applied rewrites 72.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(0.5 \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(0.5 \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right)}{\sin \left(0.5 \cdot K\right) \cdot 0 - 1 \cdot \cos \left(-0.5 \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{\left(\sin \left(0.5 \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(0.5 \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right)}{\sin \left(0.5 \cdot K\right) \cdot 0 - 1 \cdot \cos \left(-0.5 \cdot K\right)}}}\right)}^{2}} \]

   7. Applied rewrites 72.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 72.0%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.0%

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 72.0%

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

   9. Applied rewrites 72.0%

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

Alternative 3: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \left|J\right| \cdot -2\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\left|\left|J\right|\right|}\right)\right)\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\left(\sqrt{\frac{t\_0 \cdot t\_0}{\cos \left(1 \cdot K\right) \cdot 0.5 - -0.5} - -1} \cdot t\_4\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 \cdot \left(\frac{\sqrt{\frac{0.25}{\left|J\right| \cdot \left|J\right|}}}{\left|t\_1\right|} \cdot t\_1\right)\right) \cdot \left|U\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (/ (fabs U) (+ (fabs J) (fabs J))))
       (t_1 (cos (* -0.5 K)))
       (t_2 (cos (/ K 2.0)))
       (t_3
        (*
         (* (* -2.0 (fabs J)) t_2)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0)))))
       (t_4 (* (fabs J) -2.0)))
  (*
   (copysign 1.0 J)
   (if (<= t_3 (- INFINITY))
     (*
      -2.0
      (*
       (fabs J)
       (*
        (fabs U)
        (*
         (cos (* 0.5 K))
         (/
          (sqrt (/ 0.25 (+ 0.5 (* 0.5 (cos (* 2.0 (* -0.5 K)))))))
          (fabs (fabs J)))))))
     (if (<= t_3 2e+289)
       (*
        (*
         (sqrt
          (- (/ (* t_0 t_0) (- (* (cos (* 1.0 K)) 0.5) -0.5)) -1.0))
         t_4)
        (cos (* K 0.5)))
       (*
        (*
         t_4
         (* (/ (sqrt (/ 0.25 (* (fabs J) (fabs J)))) (fabs t_1)) t_1))
        (fabs U)))))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(U) / (fabs(J) + fabs(J));
	double t_1 = cos((-0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * fabs(J)) * t_2) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_4 = fabs(J) * -2.0;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -2.0 * (fabs(J) * (fabs(U) * (cos((0.5 * K)) * (sqrt((0.25 / (0.5 + (0.5 * cos((2.0 * (-0.5 * K))))))) / fabs(fabs(J))))));
	} else if (t_3 <= 2e+289) {
		tmp = (sqrt((((t_0 * t_0) / ((cos((1.0 * K)) * 0.5) - -0.5)) - -1.0)) * t_4) * cos((K * 0.5));
	} else {
		tmp = (t_4 * ((sqrt((0.25 / (fabs(J) * fabs(J)))) / fabs(t_1)) * t_1)) * fabs(U);
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.abs(U) / (Math.abs(J) + Math.abs(J));
	double t_1 = Math.cos((-0.5 * K));
	double t_2 = Math.cos((K / 2.0));
	double t_3 = ((-2.0 * Math.abs(J)) * t_2) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_2)), 2.0)));
	double t_4 = Math.abs(J) * -2.0;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (Math.abs(J) * (Math.abs(U) * (Math.cos((0.5 * K)) * (Math.sqrt((0.25 / (0.5 + (0.5 * Math.cos((2.0 * (-0.5 * K))))))) / Math.abs(Math.abs(J))))));
	} else if (t_3 <= 2e+289) {
		tmp = (Math.sqrt((((t_0 * t_0) / ((Math.cos((1.0 * K)) * 0.5) - -0.5)) - -1.0)) * t_4) * Math.cos((K * 0.5));
	} else {
		tmp = (t_4 * ((Math.sqrt((0.25 / (Math.abs(J) * Math.abs(J)))) / Math.abs(t_1)) * t_1)) * Math.abs(U);
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.fabs(U) / (math.fabs(J) + math.fabs(J))
	t_1 = math.cos((-0.5 * K))
	t_2 = math.cos((K / 2.0))
	t_3 = ((-2.0 * math.fabs(J)) * t_2) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_2)), 2.0)))
	t_4 = math.fabs(J) * -2.0
	tmp = 0
	if t_3 <= -math.inf:
		tmp = -2.0 * (math.fabs(J) * (math.fabs(U) * (math.cos((0.5 * K)) * (math.sqrt((0.25 / (0.5 + (0.5 * math.cos((2.0 * (-0.5 * K))))))) / math.fabs(math.fabs(J))))))
	elif t_3 <= 2e+289:
		tmp = (math.sqrt((((t_0 * t_0) / ((math.cos((1.0 * K)) * 0.5) - -0.5)) - -1.0)) * t_4) * math.cos((K * 0.5))
	else:
		tmp = (t_4 * ((math.sqrt((0.25 / (math.fabs(J) * math.fabs(J)))) / math.fabs(t_1)) * t_1)) * math.fabs(U)
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * abs(J)) * t_2) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_4 = Float64(abs(J) * -2.0)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(abs(J) * Float64(abs(U) * Float64(cos(Float64(0.5 * K)) * Float64(sqrt(Float64(0.25 / Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(-0.5 * K))))))) / abs(abs(J)))))));
	elseif (t_3 <= 2e+289)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(t_0 * t_0) / Float64(Float64(cos(Float64(1.0 * K)) * 0.5) - -0.5)) - -1.0)) * t_4) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(Float64(t_4 * Float64(Float64(sqrt(Float64(0.25 / Float64(abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = abs(U) / (abs(J) + abs(J));
	t_1 = cos((-0.5 * K));
	t_2 = cos((K / 2.0));
	t_3 = ((-2.0 * abs(J)) * t_2) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_2)) ^ 2.0)));
	t_4 = abs(J) * -2.0;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -2.0 * (abs(J) * (abs(U) * (cos((0.5 * K)) * (sqrt((0.25 / (0.5 + (0.5 * cos((2.0 * (-0.5 * K))))))) / abs(abs(J))))));
	elseif (t_3 <= 2e+289)
		tmp = (sqrt((((t_0 * t_0) / ((cos((1.0 * K)) * 0.5) - -0.5)) - -1.0)) * t_4) * cos((K * 0.5));
	else
		tmp = (t_4 * ((sqrt((0.25 / (abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U);
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $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 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(0.25 / N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(-0.5 * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[J], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+289], N[(N[(N[Sqrt[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[(N[Cos[N[(1.0 * K), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 * N[(N[(N[Sqrt[N[(0.25 / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 21.3%

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

   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))))) < 2.0000000000000001e289

   1. Initial program 72.2%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 72.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{\left(\sin \left(0.5 \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(0.5 \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right)}{\sin \left(0.5 \cdot K\right) \cdot 0 - 1 \cdot \cos \left(-0.5 \cdot K\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}\right)}^{2}} \]

      4. flip-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}{\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) - \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}}\right)}^{2}} \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(\frac{1}{2} \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)}{\sin \left(\frac{1}{2} \cdot K\right) \cdot 0 - 1 \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}{\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) - \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}}\right)}^{2}} \]

   5. Applied rewrites 72.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{\left(\sin \left(0.5 \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(0.5 \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right)}{\sin \left(0.5 \cdot K\right) \cdot 0 - 1 \cdot \cos \left(-0.5 \cdot K\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{\left(\sin \left(0.5 \cdot K\right) \cdot 0\right) \cdot \left(\sin \left(0.5 \cdot K\right) \cdot 0\right) - \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(1 \cdot \cos \left(-0.5 \cdot K\right)\right)}{\sin \left(0.5 \cdot K\right) \cdot 0 - 1 \cdot \cos \left(-0.5 \cdot K\right)}}}\right)}^{2}} \]

   7. Applied rewrites 72.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 72.0%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.0%

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 72.0%

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

   9. Applied rewrites 72.0%

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

   if 2.0000000000000001e289 < (*.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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 15.0%

      \[\leadsto \left(\left(J \cdot -2\right) \cdot \left(\frac{\sqrt{\frac{0.25}{J \cdot J}}}{\left|\cos \left(-0.5 \cdot K\right)\right|} \cdot \cos \left(-0.5 \cdot K\right)\right)\right) \cdot \color{blue}{U} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|J\right| \cdot -2\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \cos \left(-0.5 \cdot K\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\left|\left|J\right|\right|}\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\left(\sqrt{\frac{\left|U\right|}{\left(\left(\cos K \cdot 0.5 - -0.5\right) \cdot 4\right) \cdot \left|J\right|} \cdot \frac{\left|U\right|}{\left|J\right|} - -1} \cdot t\_3\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(\frac{\sqrt{\frac{0.25}{\left|J\right| \cdot \left|J\right|}}}{\left|t\_3\right|} \cdot t\_3\right)\right) \cdot \left|U\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (* (fabs J) -2.0))
       (t_1 (cos (/ K 2.0)))
       (t_2
        (*
         (* (* -2.0 (fabs J)) t_1)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0)))))
       (t_3 (cos (* -0.5 K))))
  (*
   (copysign 1.0 J)
   (if (<= t_2 (- INFINITY))
     (*
      -2.0
      (*
       (fabs J)
       (*
        (fabs U)
        (*
         (cos (* 0.5 K))
         (/
          (sqrt (/ 0.25 (+ 0.5 (* 0.5 (cos (* 2.0 (* -0.5 K)))))))
          (fabs (fabs J)))))))
     (if (<= t_2 2e+289)
       (*
        (*
         (sqrt
          (-
           (*
            (/ (fabs U) (* (* (- (* (cos K) 0.5) -0.5) 4.0) (fabs J)))
            (/ (fabs U) (fabs J)))
           -1.0))
         t_3)
        t_0)
       (*
        (*
         t_0
         (* (/ (sqrt (/ 0.25 (* (fabs J) (fabs J)))) (fabs t_3)) t_3))
        (fabs U)))))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(J) * -2.0;
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * fabs(J)) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double t_3 = cos((-0.5 * K));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (fabs(J) * (fabs(U) * (cos((0.5 * K)) * (sqrt((0.25 / (0.5 + (0.5 * cos((2.0 * (-0.5 * K))))))) / fabs(fabs(J))))));
	} else if (t_2 <= 2e+289) {
		tmp = (sqrt((((fabs(U) / ((((cos(K) * 0.5) - -0.5) * 4.0) * fabs(J))) * (fabs(U) / fabs(J))) - -1.0)) * t_3) * t_0;
	} else {
		tmp = (t_0 * ((sqrt((0.25 / (fabs(J) * fabs(J)))) / fabs(t_3)) * t_3)) * fabs(U);
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.abs(J) * -2.0;
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * Math.abs(J)) * t_1) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double t_3 = Math.cos((-0.5 * K));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (Math.abs(J) * (Math.abs(U) * (Math.cos((0.5 * K)) * (Math.sqrt((0.25 / (0.5 + (0.5 * Math.cos((2.0 * (-0.5 * K))))))) / Math.abs(Math.abs(J))))));
	} else if (t_2 <= 2e+289) {
		tmp = (Math.sqrt((((Math.abs(U) / ((((Math.cos(K) * 0.5) - -0.5) * 4.0) * Math.abs(J))) * (Math.abs(U) / Math.abs(J))) - -1.0)) * t_3) * t_0;
	} else {
		tmp = (t_0 * ((Math.sqrt((0.25 / (Math.abs(J) * Math.abs(J)))) / Math.abs(t_3)) * t_3)) * Math.abs(U);
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.fabs(J) * -2.0
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * math.fabs(J)) * t_1) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	t_3 = math.cos((-0.5 * K))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (math.fabs(J) * (math.fabs(U) * (math.cos((0.5 * K)) * (math.sqrt((0.25 / (0.5 + (0.5 * math.cos((2.0 * (-0.5 * K))))))) / math.fabs(math.fabs(J))))))
	elif t_2 <= 2e+289:
		tmp = (math.sqrt((((math.fabs(U) / ((((math.cos(K) * 0.5) - -0.5) * 4.0) * math.fabs(J))) * (math.fabs(U) / math.fabs(J))) - -1.0)) * t_3) * t_0
	else:
		tmp = (t_0 * ((math.sqrt((0.25 / (math.fabs(J) * math.fabs(J)))) / math.fabs(t_3)) * t_3)) * math.fabs(U)
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(abs(J) * -2.0)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * abs(J)) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	t_3 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(abs(J) * Float64(abs(U) * Float64(cos(Float64(0.5 * K)) * Float64(sqrt(Float64(0.25 / Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(-0.5 * K))))))) / abs(abs(J)))))));
	elseif (t_2 <= 2e+289)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(abs(U) / Float64(Float64(Float64(Float64(cos(K) * 0.5) - -0.5) * 4.0) * abs(J))) * Float64(abs(U) / abs(J))) - -1.0)) * t_3) * t_0);
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(sqrt(Float64(0.25 / Float64(abs(J) * abs(J)))) / abs(t_3)) * t_3)) * abs(U));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = abs(J) * -2.0;
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * abs(J)) * t_1) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	t_3 = cos((-0.5 * K));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (abs(J) * (abs(U) * (cos((0.5 * K)) * (sqrt((0.25 / (0.5 + (0.5 * cos((2.0 * (-0.5 * K))))))) / abs(abs(J))))));
	elseif (t_2 <= 2e+289)
		tmp = (sqrt((((abs(U) / ((((cos(K) * 0.5) - -0.5) * 4.0) * abs(J))) * (abs(U) / abs(J))) - -1.0)) * t_3) * t_0;
	else
		tmp = (t_0 * ((sqrt((0.25 / (abs(J) * abs(J)))) / abs(t_3)) * t_3)) * abs(U);
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $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[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(0.25 / N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(-0.5 * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[J], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+289], N[(N[(N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[(N[(N[Cos[K], $MachinePrecision] * 0.5), $MachinePrecision] - -0.5), $MachinePrecision] * 4.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 * N[(N[(N[Sqrt[N[(0.25 / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$3], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 21.3%

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

   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))))) < 2.0000000000000001e289

   1. Initial program 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.4%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 60.4%

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

   7. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 72.0%

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

   9. Applied rewrites 72.0%

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

   if 2.0000000000000001e289 < (*.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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 15.0%

      \[\leadsto \left(\left(J \cdot -2\right) \cdot \left(\frac{\sqrt{\frac{0.25}{J \cdot J}}}{\left|\cos \left(-0.5 \cdot K\right)\right|} \cdot \cos \left(-0.5 \cdot K\right)\right)\right) \cdot \color{blue}{U} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \left|J\right| \cdot -2\\ t_2 := \sqrt{\frac{\left|U\right|}{\left(\left(\left(\cos K \cdot 0.5 - -0.5\right) \cdot 4\right) \cdot \left|J\right|\right) \cdot \left|J\right|} \cdot \left|U\right| - -1}\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_3\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_3}\right)}^{2}}\\ t_5 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \frac{\sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)}}}{\left|\left|J\right|\right|}\right)\right)\right)\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-77}:\\ \;\;\;\;\left(t\_1 \cdot t\_2\right) \cdot t\_0\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{-80}:\\ \;\;\;\;\left(\sqrt{t\_5 \cdot t\_5 - -1} \cdot -2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left|J\right|\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\left(t\_0 \cdot t\_1\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \left(\frac{\sqrt{\frac{0.25}{\left|J\right| \cdot \left|J\right|}}}{\left|t\_0\right|} \cdot t\_0\right)\right) \cdot \left|U\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -0.5 K)))
       (t_1 (* (fabs J) -2.0))
       (t_2
        (sqrt
         (-
          (*
           (/
            (fabs U)
            (*
             (* (* (- (* (cos K) 0.5) -0.5) 4.0) (fabs J))
             (fabs J)))
           (fabs U))
          -1.0)))
       (t_3 (cos (/ K 2.0)))
       (t_4
        (*
         (* (* -2.0 (fabs J)) t_3)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_3)) 2.0)))))
       (t_5 (/ (fabs U) (+ (fabs J) (fabs J)))))
  (*
   (copysign 1.0 J)
   (if (<= t_4 (- INFINITY))
     (*
      -2.0
      (*
       (fabs J)
       (*
        (fabs U)
        (*
         (cos (* 0.5 K))
         (/
          (sqrt (/ 0.25 (+ 0.5 (* 0.5 (cos (* 2.0 (* -0.5 K)))))))
          (fabs (fabs J)))))))
     (if (<= t_4 -1e-77)
       (* (* t_1 t_2) t_0)
       (if (<= t_4 4e-80)
         (*
          (* (sqrt (- (* t_5 t_5) -1.0)) -2.0)
          (* (cos (* K 0.5)) (fabs J)))
         (if (<= t_4 2e+289)
           (* (* t_0 t_1) t_2)
           (*
            (*
             t_1
             (*
              (/ (sqrt (/ 0.25 (* (fabs J) (fabs J)))) (fabs t_0))
              t_0))
            (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = fabs(J) * -2.0;
	double t_2 = sqrt((((fabs(U) / (((((cos(K) * 0.5) - -0.5) * 4.0) * fabs(J)) * fabs(J))) * fabs(U)) - -1.0));
	double t_3 = cos((K / 2.0));
	double t_4 = ((-2.0 * fabs(J)) * t_3) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_3)), 2.0)));
	double t_5 = fabs(U) / (fabs(J) + fabs(J));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -2.0 * (fabs(J) * (fabs(U) * (cos((0.5 * K)) * (sqrt((0.25 / (0.5 + (0.5 * cos((2.0 * (-0.5 * K))))))) / fabs(fabs(J))))));
	} else if (t_4 <= -1e-77) {
		tmp = (t_1 * t_2) * t_0;
	} else if (t_4 <= 4e-80) {
		tmp = (sqrt(((t_5 * t_5) - -1.0)) * -2.0) * (cos((K * 0.5)) * fabs(J));
	} else if (t_4 <= 2e+289) {
		tmp = (t_0 * t_1) * t_2;
	} else {
		tmp = (t_1 * ((sqrt((0.25 / (fabs(J) * fabs(J)))) / fabs(t_0)) * t_0)) * fabs(U);
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.abs(J) * -2.0;
	double t_2 = Math.sqrt((((Math.abs(U) / (((((Math.cos(K) * 0.5) - -0.5) * 4.0) * Math.abs(J)) * Math.abs(J))) * Math.abs(U)) - -1.0));
	double t_3 = Math.cos((K / 2.0));
	double t_4 = ((-2.0 * Math.abs(J)) * t_3) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_3)), 2.0)));
	double t_5 = Math.abs(U) / (Math.abs(J) + Math.abs(J));
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (Math.abs(J) * (Math.abs(U) * (Math.cos((0.5 * K)) * (Math.sqrt((0.25 / (0.5 + (0.5 * Math.cos((2.0 * (-0.5 * K))))))) / Math.abs(Math.abs(J))))));
	} else if (t_4 <= -1e-77) {
		tmp = (t_1 * t_2) * t_0;
	} else if (t_4 <= 4e-80) {
		tmp = (Math.sqrt(((t_5 * t_5) - -1.0)) * -2.0) * (Math.cos((K * 0.5)) * Math.abs(J));
	} else if (t_4 <= 2e+289) {
		tmp = (t_0 * t_1) * t_2;
	} else {
		tmp = (t_1 * ((Math.sqrt((0.25 / (Math.abs(J) * Math.abs(J)))) / Math.abs(t_0)) * t_0)) * Math.abs(U);
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.fabs(J) * -2.0
	t_2 = math.sqrt((((math.fabs(U) / (((((math.cos(K) * 0.5) - -0.5) * 4.0) * math.fabs(J)) * math.fabs(J))) * math.fabs(U)) - -1.0))
	t_3 = math.cos((K / 2.0))
	t_4 = ((-2.0 * math.fabs(J)) * t_3) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_3)), 2.0)))
	t_5 = math.fabs(U) / (math.fabs(J) + math.fabs(J))
	tmp = 0
	if t_4 <= -math.inf:
		tmp = -2.0 * (math.fabs(J) * (math.fabs(U) * (math.cos((0.5 * K)) * (math.sqrt((0.25 / (0.5 + (0.5 * math.cos((2.0 * (-0.5 * K))))))) / math.fabs(math.fabs(J))))))
	elif t_4 <= -1e-77:
		tmp = (t_1 * t_2) * t_0
	elif t_4 <= 4e-80:
		tmp = (math.sqrt(((t_5 * t_5) - -1.0)) * -2.0) * (math.cos((K * 0.5)) * math.fabs(J))
	elif t_4 <= 2e+289:
		tmp = (t_0 * t_1) * t_2
	else:
		tmp = (t_1 * ((math.sqrt((0.25 / (math.fabs(J) * math.fabs(J)))) / math.fabs(t_0)) * t_0)) * math.fabs(U)
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(abs(J) * -2.0)
	t_2 = sqrt(Float64(Float64(Float64(abs(U) / Float64(Float64(Float64(Float64(Float64(cos(K) * 0.5) - -0.5) * 4.0) * abs(J)) * abs(J))) * abs(U)) - -1.0))
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(Float64(-2.0 * abs(J)) * t_3) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_3)) ^ 2.0))))
	t_5 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(abs(J) * Float64(abs(U) * Float64(cos(Float64(0.5 * K)) * Float64(sqrt(Float64(0.25 / Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(-0.5 * K))))))) / abs(abs(J)))))));
	elseif (t_4 <= -1e-77)
		tmp = Float64(Float64(t_1 * t_2) * t_0);
	elseif (t_4 <= 4e-80)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_5 * t_5) - -1.0)) * -2.0) * Float64(cos(Float64(K * 0.5)) * abs(J)));
	elseif (t_4 <= 2e+289)
		tmp = Float64(Float64(t_0 * t_1) * t_2);
	else
		tmp = Float64(Float64(t_1 * Float64(Float64(sqrt(Float64(0.25 / Float64(abs(J) * abs(J)))) / abs(t_0)) * t_0)) * abs(U));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K));
	t_1 = abs(J) * -2.0;
	t_2 = sqrt((((abs(U) / (((((cos(K) * 0.5) - -0.5) * 4.0) * abs(J)) * abs(J))) * abs(U)) - -1.0));
	t_3 = cos((K / 2.0));
	t_4 = ((-2.0 * abs(J)) * t_3) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_3)) ^ 2.0)));
	t_5 = abs(U) / (abs(J) + abs(J));
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = -2.0 * (abs(J) * (abs(U) * (cos((0.5 * K)) * (sqrt((0.25 / (0.5 + (0.5 * cos((2.0 * (-0.5 * K))))))) / abs(abs(J))))));
	elseif (t_4 <= -1e-77)
		tmp = (t_1 * t_2) * t_0;
	elseif (t_4 <= 4e-80)
		tmp = (sqrt(((t_5 * t_5) - -1.0)) * -2.0) * (cos((K * 0.5)) * abs(J));
	elseif (t_4 <= 2e+289)
		tmp = (t_0 * t_1) * t_2;
	else
		tmp = (t_1 * ((sqrt((0.25 / (abs(J) * abs(J)))) / abs(t_0)) * t_0)) * abs(U);
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[(N[(N[(N[Cos[K], $MachinePrecision] * 0.5), $MachinePrecision] - -0.5), $MachinePrecision] * 4.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] * N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(0.25 / N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(-0.5 * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[J], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-77], N[(N[(t$95$1 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$4, 4e-80], N[(N[(N[Sqrt[N[(N[(t$95$5 * t$95$5), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+289], N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(t$95$1 * N[(N[(N[Sqrt[N[(0.25 / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 21.3%

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

   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))))) < -9.9999999999999993e-78

   1. Initial program 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.4%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 60.4%

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

   7. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. lift-cos.f64 N/A

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

   9. Applied rewrites 60.4%

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

   if -9.9999999999999993e-78 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999998e-80

   1. Initial program 72.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 84.2%

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

      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 84.2%

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

      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 84.2%

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

   3. Applied rewrites 84.2%

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

   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 70.5%

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

   6. Applied rewrites 70.5%

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

   8. Applied rewrites 63.4%

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

   if 3.9999999999999998e-80 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e289

   1. Initial program 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.4%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 60.4%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 60.4%

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

   if 2.0000000000000001e289 < (*.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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 15.0%

      \[\leadsto \left(\left(J \cdot -2\right) \cdot \left(\frac{\sqrt{\frac{0.25}{J \cdot J}}}{\left|\cos \left(-0.5 \cdot K\right)\right|} \cdot \cos \left(-0.5 \cdot K\right)\right)\right) \cdot \color{blue}{U} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \left|J\right| \cdot -2\\ t_5 := \left(t\_4 \cdot \left(\frac{\sqrt{\frac{0.25}{\left|J\right| \cdot \left|J\right|}}}{\left|t\_1\right|} \cdot t\_1\right)\right) \cdot \left|U\right|\\ t_6 := \sqrt{\frac{\left|U\right|}{\left(\left(\left(\cos K \cdot 0.5 - -0.5\right) \cdot 4\right) \cdot \left|J\right|\right) \cdot \left|J\right|} \cdot \left|U\right| - -1}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-77}:\\ \;\;\;\;\left(t\_4 \cdot t\_6\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-80}:\\ \;\;\;\;\left(\sqrt{t\_0 \cdot t\_0 - -1} \cdot -2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left|J\right|\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\left(t\_1 \cdot t\_4\right) \cdot t\_6\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (/ (fabs U) (+ (fabs J) (fabs J))))
       (t_1 (cos (* -0.5 K)))
       (t_2 (cos (/ K 2.0)))
       (t_3
        (*
         (* (* -2.0 (fabs J)) t_2)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0)))))
       (t_4 (* (fabs J) -2.0))
       (t_5
        (*
         (*
          t_4
          (*
           (/ (sqrt (/ 0.25 (* (fabs J) (fabs J)))) (fabs t_1))
           t_1))
         (fabs U)))
       (t_6
        (sqrt
         (-
          (*
           (/
            (fabs U)
            (*
             (* (* (- (* (cos K) 0.5) -0.5) 4.0) (fabs J))
             (fabs J)))
           (fabs U))
          -1.0))))
  (*
   (copysign 1.0 J)
   (if (<= t_3 (- INFINITY))
     t_5
     (if (<= t_3 -1e-77)
       (* (* t_4 t_6) t_1)
       (if (<= t_3 4e-80)
         (*
          (* (sqrt (- (* t_0 t_0) -1.0)) -2.0)
          (* (cos (* K 0.5)) (fabs J)))
         (if (<= t_3 2e+289) (* (* t_1 t_4) t_6) t_5)))))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(U) / (fabs(J) + fabs(J));
	double t_1 = cos((-0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * fabs(J)) * t_2) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_4 = fabs(J) * -2.0;
	double t_5 = (t_4 * ((sqrt((0.25 / (fabs(J) * fabs(J)))) / fabs(t_1)) * t_1)) * fabs(U);
	double t_6 = sqrt((((fabs(U) / (((((cos(K) * 0.5) - -0.5) * 4.0) * fabs(J)) * fabs(J))) * fabs(U)) - -1.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_3 <= -1e-77) {
		tmp = (t_4 * t_6) * t_1;
	} else if (t_3 <= 4e-80) {
		tmp = (sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (cos((K * 0.5)) * fabs(J));
	} else if (t_3 <= 2e+289) {
		tmp = (t_1 * t_4) * t_6;
	} else {
		tmp = t_5;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.abs(U) / (Math.abs(J) + Math.abs(J));
	double t_1 = Math.cos((-0.5 * K));
	double t_2 = Math.cos((K / 2.0));
	double t_3 = ((-2.0 * Math.abs(J)) * t_2) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_2)), 2.0)));
	double t_4 = Math.abs(J) * -2.0;
	double t_5 = (t_4 * ((Math.sqrt((0.25 / (Math.abs(J) * Math.abs(J)))) / Math.abs(t_1)) * t_1)) * Math.abs(U);
	double t_6 = Math.sqrt((((Math.abs(U) / (((((Math.cos(K) * 0.5) - -0.5) * 4.0) * Math.abs(J)) * Math.abs(J))) * Math.abs(U)) - -1.0));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_5;
	} else if (t_3 <= -1e-77) {
		tmp = (t_4 * t_6) * t_1;
	} else if (t_3 <= 4e-80) {
		tmp = (Math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (Math.cos((K * 0.5)) * Math.abs(J));
	} else if (t_3 <= 2e+289) {
		tmp = (t_1 * t_4) * t_6;
	} else {
		tmp = t_5;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.fabs(U) / (math.fabs(J) + math.fabs(J))
	t_1 = math.cos((-0.5 * K))
	t_2 = math.cos((K / 2.0))
	t_3 = ((-2.0 * math.fabs(J)) * t_2) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_2)), 2.0)))
	t_4 = math.fabs(J) * -2.0
	t_5 = (t_4 * ((math.sqrt((0.25 / (math.fabs(J) * math.fabs(J)))) / math.fabs(t_1)) * t_1)) * math.fabs(U)
	t_6 = math.sqrt((((math.fabs(U) / (((((math.cos(K) * 0.5) - -0.5) * 4.0) * math.fabs(J)) * math.fabs(J))) * math.fabs(U)) - -1.0))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_5
	elif t_3 <= -1e-77:
		tmp = (t_4 * t_6) * t_1
	elif t_3 <= 4e-80:
		tmp = (math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (math.cos((K * 0.5)) * math.fabs(J))
	elif t_3 <= 2e+289:
		tmp = (t_1 * t_4) * t_6
	else:
		tmp = t_5
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * abs(J)) * t_2) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_4 = Float64(abs(J) * -2.0)
	t_5 = Float64(Float64(t_4 * Float64(Float64(sqrt(Float64(0.25 / Float64(abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U))
	t_6 = sqrt(Float64(Float64(Float64(abs(U) / Float64(Float64(Float64(Float64(Float64(cos(K) * 0.5) - -0.5) * 4.0) * abs(J)) * abs(J))) * abs(U)) - -1.0))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_3 <= -1e-77)
		tmp = Float64(Float64(t_4 * t_6) * t_1);
	elseif (t_3 <= 4e-80)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_0 * t_0) - -1.0)) * -2.0) * Float64(cos(Float64(K * 0.5)) * abs(J)));
	elseif (t_3 <= 2e+289)
		tmp = Float64(Float64(t_1 * t_4) * t_6);
	else
		tmp = t_5;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = abs(U) / (abs(J) + abs(J));
	t_1 = cos((-0.5 * K));
	t_2 = cos((K / 2.0));
	t_3 = ((-2.0 * abs(J)) * t_2) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_2)) ^ 2.0)));
	t_4 = abs(J) * -2.0;
	t_5 = (t_4 * ((sqrt((0.25 / (abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U);
	t_6 = sqrt((((abs(U) / (((((cos(K) * 0.5) - -0.5) * 4.0) * abs(J)) * abs(J))) * abs(U)) - -1.0));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_5;
	elseif (t_3 <= -1e-77)
		tmp = (t_4 * t_6) * t_1;
	elseif (t_3 <= 4e-80)
		tmp = (sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (cos((K * 0.5)) * abs(J));
	elseif (t_3 <= 2e+289)
		tmp = (t_1 * t_4) * t_6;
	else
		tmp = t_5;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $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 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * N[(N[(N[Sqrt[N[(0.25 / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[(N[(N[(N[Cos[K], $MachinePrecision] * 0.5), $MachinePrecision] - -0.5), $MachinePrecision] * 4.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], t$95$5, If[LessEqual[t$95$3, -1e-77], N[(N[(t$95$4 * t$95$6), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 4e-80], N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+289], N[(N[(t$95$1 * t$95$4), $MachinePrecision] * t$95$6), $MachinePrecision], t$95$5]]]]), $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 or 2.0000000000000001e289 < (*.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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 15.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.9999999999999993e-78

   1. Initial program 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.4%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 60.4%

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

   7. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. lift-cos.f64 N/A

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

   9. Applied rewrites 60.4%

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

   if -9.9999999999999993e-78 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999998e-80

   1. Initial program 72.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 84.2%

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

      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 84.2%

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

      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 84.2%

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

   3. Applied rewrites 84.2%

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

   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 70.5%

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

   6. Applied rewrites 70.5%

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

   8. Applied rewrites 63.4%

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

   if 3.9999999999999998e-80 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e289

   1. Initial program 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.4%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 60.4%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 60.4%

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

Alternative 7: 77.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos K \cdot 0.5 - -0.5\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_3\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_3}\right)}^{2}}\\ t_5 := \left|J\right| \cdot -2\\ t_6 := \left(t\_5 \cdot \left(\frac{\sqrt{\frac{0.25}{\left|J\right| \cdot \left|J\right|}}}{\left|t\_1\right|} \cdot t\_1\right)\right) \cdot \left|U\right|\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-77}:\\ \;\;\;\;\left(\sqrt{\frac{\left|U\right|}{\left(\left(4 \cdot t\_2\right) \cdot \left|J\right|\right) \cdot \left|J\right|} \cdot \left|U\right| - -1} \cdot t\_1\right) \cdot t\_5\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{-80}:\\ \;\;\;\;\left(\sqrt{t\_0 \cdot t\_0 - -1} \cdot -2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left|J\right|\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\left(t\_1 \cdot t\_5\right) \cdot \sqrt{\frac{\left|U\right|}{\left(\left(t\_2 \cdot 4\right) \cdot \left|J\right|\right) \cdot \left|J\right|} \cdot \left|U\right| - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (/ (fabs U) (+ (fabs J) (fabs J))))
       (t_1 (cos (* -0.5 K)))
       (t_2 (- (* (cos K) 0.5) -0.5))
       (t_3 (cos (/ K 2.0)))
       (t_4
        (*
         (* (* -2.0 (fabs J)) t_3)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_3)) 2.0)))))
       (t_5 (* (fabs J) -2.0))
       (t_6
        (*
         (*
          t_5
          (*
           (/ (sqrt (/ 0.25 (* (fabs J) (fabs J)))) (fabs t_1))
           t_1))
         (fabs U))))
  (*
   (copysign 1.0 J)
   (if (<= t_4 (- INFINITY))
     t_6
     (if (<= t_4 -1e-77)
       (*
        (*
         (sqrt
          (-
           (*
            (/ (fabs U) (* (* (* 4.0 t_2) (fabs J)) (fabs J)))
            (fabs U))
           -1.0))
         t_1)
        t_5)
       (if (<= t_4 4e-80)
         (*
          (* (sqrt (- (* t_0 t_0) -1.0)) -2.0)
          (* (cos (* K 0.5)) (fabs J)))
         (if (<= t_4 2e+289)
           (*
            (* t_1 t_5)
            (sqrt
             (-
              (*
               (/ (fabs U) (* (* (* t_2 4.0) (fabs J)) (fabs J)))
               (fabs U))
              -1.0)))
           t_6)))))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(U) / (fabs(J) + fabs(J));
	double t_1 = cos((-0.5 * K));
	double t_2 = (cos(K) * 0.5) - -0.5;
	double t_3 = cos((K / 2.0));
	double t_4 = ((-2.0 * fabs(J)) * t_3) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_3)), 2.0)));
	double t_5 = fabs(J) * -2.0;
	double t_6 = (t_5 * ((sqrt((0.25 / (fabs(J) * fabs(J)))) / fabs(t_1)) * t_1)) * fabs(U);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_6;
	} else if (t_4 <= -1e-77) {
		tmp = (sqrt((((fabs(U) / (((4.0 * t_2) * fabs(J)) * fabs(J))) * fabs(U)) - -1.0)) * t_1) * t_5;
	} else if (t_4 <= 4e-80) {
		tmp = (sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (cos((K * 0.5)) * fabs(J));
	} else if (t_4 <= 2e+289) {
		tmp = (t_1 * t_5) * sqrt((((fabs(U) / (((t_2 * 4.0) * fabs(J)) * fabs(J))) * fabs(U)) - -1.0));
	} else {
		tmp = t_6;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.abs(U) / (Math.abs(J) + Math.abs(J));
	double t_1 = Math.cos((-0.5 * K));
	double t_2 = (Math.cos(K) * 0.5) - -0.5;
	double t_3 = Math.cos((K / 2.0));
	double t_4 = ((-2.0 * Math.abs(J)) * t_3) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_3)), 2.0)));
	double t_5 = Math.abs(J) * -2.0;
	double t_6 = (t_5 * ((Math.sqrt((0.25 / (Math.abs(J) * Math.abs(J)))) / Math.abs(t_1)) * t_1)) * Math.abs(U);
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = t_6;
	} else if (t_4 <= -1e-77) {
		tmp = (Math.sqrt((((Math.abs(U) / (((4.0 * t_2) * Math.abs(J)) * Math.abs(J))) * Math.abs(U)) - -1.0)) * t_1) * t_5;
	} else if (t_4 <= 4e-80) {
		tmp = (Math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (Math.cos((K * 0.5)) * Math.abs(J));
	} else if (t_4 <= 2e+289) {
		tmp = (t_1 * t_5) * Math.sqrt((((Math.abs(U) / (((t_2 * 4.0) * Math.abs(J)) * Math.abs(J))) * Math.abs(U)) - -1.0));
	} else {
		tmp = t_6;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.fabs(U) / (math.fabs(J) + math.fabs(J))
	t_1 = math.cos((-0.5 * K))
	t_2 = (math.cos(K) * 0.5) - -0.5
	t_3 = math.cos((K / 2.0))
	t_4 = ((-2.0 * math.fabs(J)) * t_3) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_3)), 2.0)))
	t_5 = math.fabs(J) * -2.0
	t_6 = (t_5 * ((math.sqrt((0.25 / (math.fabs(J) * math.fabs(J)))) / math.fabs(t_1)) * t_1)) * math.fabs(U)
	tmp = 0
	if t_4 <= -math.inf:
		tmp = t_6
	elif t_4 <= -1e-77:
		tmp = (math.sqrt((((math.fabs(U) / (((4.0 * t_2) * math.fabs(J)) * math.fabs(J))) * math.fabs(U)) - -1.0)) * t_1) * t_5
	elif t_4 <= 4e-80:
		tmp = (math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (math.cos((K * 0.5)) * math.fabs(J))
	elif t_4 <= 2e+289:
		tmp = (t_1 * t_5) * math.sqrt((((math.fabs(U) / (((t_2 * 4.0) * math.fabs(J)) * math.fabs(J))) * math.fabs(U)) - -1.0))
	else:
		tmp = t_6
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = Float64(Float64(cos(K) * 0.5) - -0.5)
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(Float64(-2.0 * abs(J)) * t_3) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_3)) ^ 2.0))))
	t_5 = Float64(abs(J) * -2.0)
	t_6 = Float64(Float64(t_5 * Float64(Float64(sqrt(Float64(0.25 / Float64(abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_6;
	elseif (t_4 <= -1e-77)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(abs(U) / Float64(Float64(Float64(4.0 * t_2) * abs(J)) * abs(J))) * abs(U)) - -1.0)) * t_1) * t_5);
	elseif (t_4 <= 4e-80)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_0 * t_0) - -1.0)) * -2.0) * Float64(cos(Float64(K * 0.5)) * abs(J)));
	elseif (t_4 <= 2e+289)
		tmp = Float64(Float64(t_1 * t_5) * sqrt(Float64(Float64(Float64(abs(U) / Float64(Float64(Float64(t_2 * 4.0) * abs(J)) * abs(J))) * abs(U)) - -1.0)));
	else
		tmp = t_6;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = abs(U) / (abs(J) + abs(J));
	t_1 = cos((-0.5 * K));
	t_2 = (cos(K) * 0.5) - -0.5;
	t_3 = cos((K / 2.0));
	t_4 = ((-2.0 * abs(J)) * t_3) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_3)) ^ 2.0)));
	t_5 = abs(J) * -2.0;
	t_6 = (t_5 * ((sqrt((0.25 / (abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U);
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = t_6;
	elseif (t_4 <= -1e-77)
		tmp = (sqrt((((abs(U) / (((4.0 * t_2) * abs(J)) * abs(J))) * abs(U)) - -1.0)) * t_1) * t_5;
	elseif (t_4 <= 4e-80)
		tmp = (sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (cos((K * 0.5)) * abs(J));
	elseif (t_4 <= 2e+289)
		tmp = (t_1 * t_5) * sqrt((((abs(U) / (((t_2 * 4.0) * abs(J)) * abs(J))) * abs(U)) - -1.0));
	else
		tmp = t_6;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[K], $MachinePrecision] * 0.5), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$5 * N[(N[(N[Sqrt[N[(0.25 / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], t$95$6, If[LessEqual[t$95$4, -1e-77], N[(N[(N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[(4.0 * t$95$2), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$4, 4e-80], N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+289], N[(N[(t$95$1 * t$95$5), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[(t$95$2 * 4.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$6]]]]), $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 or 2.0000000000000001e289 < (*.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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 15.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.9999999999999993e-78

   1. Initial program 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.4%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 60.4%

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

   7. Applied rewrites 60.4%

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

   if -9.9999999999999993e-78 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999998e-80

   1. Initial program 72.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 84.2%

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

      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 84.2%

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

      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 84.2%

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

   3. Applied rewrites 84.2%

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

   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 70.5%

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

   6. Applied rewrites 70.5%

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

   8. Applied rewrites 63.4%

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

   if 3.9999999999999998e-80 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e289

   1. Initial program 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.4%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 60.4%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 60.4%

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

Alternative 8: 77.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \left|J\right| \cdot -2\\ t_5 := \left(t\_4 \cdot \left(\frac{\sqrt{\frac{0.25}{\left|J\right| \cdot \left|J\right|}}}{\left|t\_1\right|} \cdot t\_1\right)\right) \cdot \left|U\right|\\ t_6 := \left(t\_1 \cdot t\_4\right) \cdot \sqrt{\frac{\left|U\right|}{\left(\left(\left(\cos K \cdot 0.5 - -0.5\right) \cdot 4\right) \cdot \left|J\right|\right) \cdot \left|J\right|} \cdot \left|U\right| - -1}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-77}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-80}:\\ \;\;\;\;\left(\sqrt{t\_0 \cdot t\_0 - -1} \cdot -2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left|J\right|\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (/ (fabs U) (+ (fabs J) (fabs J))))
       (t_1 (cos (* -0.5 K)))
       (t_2 (cos (/ K 2.0)))
       (t_3
        (*
         (* (* -2.0 (fabs J)) t_2)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0)))))
       (t_4 (* (fabs J) -2.0))
       (t_5
        (*
         (*
          t_4
          (*
           (/ (sqrt (/ 0.25 (* (fabs J) (fabs J)))) (fabs t_1))
           t_1))
         (fabs U)))
       (t_6
        (*
         (* t_1 t_4)
         (sqrt
          (-
           (*
            (/
             (fabs U)
             (*
              (* (* (- (* (cos K) 0.5) -0.5) 4.0) (fabs J))
              (fabs J)))
            (fabs U))
           -1.0)))))
  (*
   (copysign 1.0 J)
   (if (<= t_3 (- INFINITY))
     t_5
     (if (<= t_3 -1e-77)
       t_6
       (if (<= t_3 4e-80)
         (*
          (* (sqrt (- (* t_0 t_0) -1.0)) -2.0)
          (* (cos (* K 0.5)) (fabs J)))
         (if (<= t_3 2e+289) t_6 t_5)))))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(U) / (fabs(J) + fabs(J));
	double t_1 = cos((-0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * fabs(J)) * t_2) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_4 = fabs(J) * -2.0;
	double t_5 = (t_4 * ((sqrt((0.25 / (fabs(J) * fabs(J)))) / fabs(t_1)) * t_1)) * fabs(U);
	double t_6 = (t_1 * t_4) * sqrt((((fabs(U) / (((((cos(K) * 0.5) - -0.5) * 4.0) * fabs(J)) * fabs(J))) * fabs(U)) - -1.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_3 <= -1e-77) {
		tmp = t_6;
	} else if (t_3 <= 4e-80) {
		tmp = (sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (cos((K * 0.5)) * fabs(J));
	} else if (t_3 <= 2e+289) {
		tmp = t_6;
	} else {
		tmp = t_5;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.abs(U) / (Math.abs(J) + Math.abs(J));
	double t_1 = Math.cos((-0.5 * K));
	double t_2 = Math.cos((K / 2.0));
	double t_3 = ((-2.0 * Math.abs(J)) * t_2) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_2)), 2.0)));
	double t_4 = Math.abs(J) * -2.0;
	double t_5 = (t_4 * ((Math.sqrt((0.25 / (Math.abs(J) * Math.abs(J)))) / Math.abs(t_1)) * t_1)) * Math.abs(U);
	double t_6 = (t_1 * t_4) * Math.sqrt((((Math.abs(U) / (((((Math.cos(K) * 0.5) - -0.5) * 4.0) * Math.abs(J)) * Math.abs(J))) * Math.abs(U)) - -1.0));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_5;
	} else if (t_3 <= -1e-77) {
		tmp = t_6;
	} else if (t_3 <= 4e-80) {
		tmp = (Math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (Math.cos((K * 0.5)) * Math.abs(J));
	} else if (t_3 <= 2e+289) {
		tmp = t_6;
	} else {
		tmp = t_5;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.fabs(U) / (math.fabs(J) + math.fabs(J))
	t_1 = math.cos((-0.5 * K))
	t_2 = math.cos((K / 2.0))
	t_3 = ((-2.0 * math.fabs(J)) * t_2) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_2)), 2.0)))
	t_4 = math.fabs(J) * -2.0
	t_5 = (t_4 * ((math.sqrt((0.25 / (math.fabs(J) * math.fabs(J)))) / math.fabs(t_1)) * t_1)) * math.fabs(U)
	t_6 = (t_1 * t_4) * math.sqrt((((math.fabs(U) / (((((math.cos(K) * 0.5) - -0.5) * 4.0) * math.fabs(J)) * math.fabs(J))) * math.fabs(U)) - -1.0))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_5
	elif t_3 <= -1e-77:
		tmp = t_6
	elif t_3 <= 4e-80:
		tmp = (math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (math.cos((K * 0.5)) * math.fabs(J))
	elif t_3 <= 2e+289:
		tmp = t_6
	else:
		tmp = t_5
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * abs(J)) * t_2) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_4 = Float64(abs(J) * -2.0)
	t_5 = Float64(Float64(t_4 * Float64(Float64(sqrt(Float64(0.25 / Float64(abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U))
	t_6 = Float64(Float64(t_1 * t_4) * sqrt(Float64(Float64(Float64(abs(U) / Float64(Float64(Float64(Float64(Float64(cos(K) * 0.5) - -0.5) * 4.0) * abs(J)) * abs(J))) * abs(U)) - -1.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_3 <= -1e-77)
		tmp = t_6;
	elseif (t_3 <= 4e-80)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_0 * t_0) - -1.0)) * -2.0) * Float64(cos(Float64(K * 0.5)) * abs(J)));
	elseif (t_3 <= 2e+289)
		tmp = t_6;
	else
		tmp = t_5;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = abs(U) / (abs(J) + abs(J));
	t_1 = cos((-0.5 * K));
	t_2 = cos((K / 2.0));
	t_3 = ((-2.0 * abs(J)) * t_2) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_2)) ^ 2.0)));
	t_4 = abs(J) * -2.0;
	t_5 = (t_4 * ((sqrt((0.25 / (abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U);
	t_6 = (t_1 * t_4) * sqrt((((abs(U) / (((((cos(K) * 0.5) - -0.5) * 4.0) * abs(J)) * abs(J))) * abs(U)) - -1.0));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_5;
	elseif (t_3 <= -1e-77)
		tmp = t_6;
	elseif (t_3 <= 4e-80)
		tmp = (sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (cos((K * 0.5)) * abs(J));
	elseif (t_3 <= 2e+289)
		tmp = t_6;
	else
		tmp = t_5;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $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 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * N[(N[(N[Sqrt[N[(0.25 / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$1 * t$95$4), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[(N[(N[(N[Cos[K], $MachinePrecision] * 0.5), $MachinePrecision] - -0.5), $MachinePrecision] * 4.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], t$95$5, If[LessEqual[t$95$3, -1e-77], t$95$6, If[LessEqual[t$95$3, 4e-80], N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+289], t$95$6, t$95$5]]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000001e289 < (*.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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 15.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.9999999999999993e-78 or 3.9999999999999998e-80 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e289

   1. Initial program 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.4%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 60.4%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 60.4%

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

   if -9.9999999999999993e-78 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999998e-80

   1. Initial program 72.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 84.2%

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

      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 84.2%

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

      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 84.2%

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

   3. Applied rewrites 84.2%

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

   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 70.5%

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

   6. Applied rewrites 70.5%

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

   8. Applied rewrites 63.4%

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

Alternative 9: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \left(\left(\left|J\right| \cdot -2\right) \cdot \left(\frac{\sqrt{\frac{0.25}{\left|J\right| \cdot \left|J\right|}}}{\left|t\_1\right|} \cdot t\_1\right)\right) \cdot \left|U\right|\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_3\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_3}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\left(\sqrt{t\_0 \cdot t\_0 - -1} \cdot -2\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left|J\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (/ (fabs U) (+ (fabs J) (fabs J))))
       (t_1 (cos (* -0.5 K)))
       (t_2
        (*
         (*
          (* (fabs J) -2.0)
          (*
           (/ (sqrt (/ 0.25 (* (fabs J) (fabs J)))) (fabs t_1))
           t_1))
         (fabs U)))
       (t_3 (cos (/ K 2.0)))
       (t_4
        (*
         (* (* -2.0 (fabs J)) t_3)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_3)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_4 (- INFINITY))
     t_2
     (if (<= t_4 2e+289)
       (*
        (* (sqrt (- (* t_0 t_0) -1.0)) -2.0)
        (* (cos (* K 0.5)) (fabs J)))
       t_2)))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(U) / (fabs(J) + fabs(J));
	double t_1 = cos((-0.5 * K));
	double t_2 = ((fabs(J) * -2.0) * ((sqrt((0.25 / (fabs(J) * fabs(J)))) / fabs(t_1)) * t_1)) * fabs(U);
	double t_3 = cos((K / 2.0));
	double t_4 = ((-2.0 * fabs(J)) * t_3) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_3)), 2.0)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_4 <= 2e+289) {
		tmp = (sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (cos((K * 0.5)) * fabs(J));
	} else {
		tmp = t_2;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.abs(U) / (Math.abs(J) + Math.abs(J));
	double t_1 = Math.cos((-0.5 * K));
	double t_2 = ((Math.abs(J) * -2.0) * ((Math.sqrt((0.25 / (Math.abs(J) * Math.abs(J)))) / Math.abs(t_1)) * t_1)) * Math.abs(U);
	double t_3 = Math.cos((K / 2.0));
	double t_4 = ((-2.0 * Math.abs(J)) * t_3) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_3)), 2.0)));
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_4 <= 2e+289) {
		tmp = (Math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (Math.cos((K * 0.5)) * Math.abs(J));
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.fabs(U) / (math.fabs(J) + math.fabs(J))
	t_1 = math.cos((-0.5 * K))
	t_2 = ((math.fabs(J) * -2.0) * ((math.sqrt((0.25 / (math.fabs(J) * math.fabs(J)))) / math.fabs(t_1)) * t_1)) * math.fabs(U)
	t_3 = math.cos((K / 2.0))
	t_4 = ((-2.0 * math.fabs(J)) * t_3) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_3)), 2.0)))
	tmp = 0
	if t_4 <= -math.inf:
		tmp = t_2
	elif t_4 <= 2e+289:
		tmp = (math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (math.cos((K * 0.5)) * math.fabs(J))
	else:
		tmp = t_2
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = Float64(Float64(Float64(abs(J) * -2.0) * Float64(Float64(sqrt(Float64(0.25 / Float64(abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U))
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(Float64(-2.0 * abs(J)) * t_3) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_3)) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_4 <= 2e+289)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_0 * t_0) - -1.0)) * -2.0) * Float64(cos(Float64(K * 0.5)) * abs(J)));
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = abs(U) / (abs(J) + abs(J));
	t_1 = cos((-0.5 * K));
	t_2 = ((abs(J) * -2.0) * ((sqrt((0.25 / (abs(J) * abs(J)))) / abs(t_1)) * t_1)) * abs(U);
	t_3 = cos((K / 2.0));
	t_4 = ((-2.0 * abs(J)) * t_3) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_3)) ^ 2.0)));
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = t_2;
	elseif (t_4 <= 2e+289)
		tmp = (sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (cos((K * 0.5)) * abs(J));
	else
		tmp = t_2;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[(N[Sqrt[N[(0.25 / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], t$95$2, If[LessEqual[t$95$4, 2e+289], N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $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 2.0000000000000001e289 < (*.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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   6. Applied rewrites 15.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e289

   1. Initial program 72.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 84.2%

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

      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 84.2%

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

      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 84.2%

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

   3. Applied rewrites 84.2%

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

   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 70.5%

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

   6. Applied rewrites 70.5%

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

   8. Applied rewrites 63.4%

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

Alternative 10: 70.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (/ (fabs U) (+ (fabs J) (fabs J)))) (t_1 (cos (/ K 2.0))))
  (*
   (copysign 1.0 J)
   (if (<=
        (*
         (* (* -2.0 (fabs J)) t_1)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0))))
        (- INFINITY))
     (*
      -2.0
      (* (fabs J) (* (fabs U) (sqrt (/ 0.25 (pow (fabs J) 2.0))))))
     (*
      (* (sqrt (- (* t_0 t_0) -1.0)) -2.0)
      (* (cos (* K 0.5)) (fabs J)))))))

C

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

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.abs(U) / (Math.abs(J) + Math.abs(J));
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * Math.abs(J)) * t_1) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)))) <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (Math.abs(J) * (Math.abs(U) * Math.sqrt((0.25 / Math.pow(Math.abs(J), 2.0)))));
	} else {
		tmp = (Math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (Math.cos((K * 0.5)) * Math.abs(J));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.fabs(U) / (math.fabs(J) + math.fabs(J))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * math.fabs(J)) * t_1) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))) <= -math.inf:
		tmp = -2.0 * (math.fabs(J) * (math.fabs(U) * math.sqrt((0.25 / math.pow(math.fabs(J), 2.0)))))
	else:
		tmp = (math.sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (math.cos((K * 0.5)) * math.fabs(J))
	return math.copysign(1.0, J) * tmp

Julia

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

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = abs(U) / (abs(J) + abs(J));
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * abs(J)) * t_1) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)))) <= -Inf)
		tmp = -2.0 * (abs(J) * (abs(U) * sqrt((0.25 / (abs(J) ^ 2.0)))));
	else
		tmp = (sqrt(((t_0 * t_0) - -1.0)) * -2.0) * (cos((K * 0.5)) * abs(J));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] * N[Sqrt[N[(0.25 / N[Power[N[Abs[J], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 13.3%

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

   7. Applied rewrites 13.3%

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

   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 72.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 84.2%

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

      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 84.2%

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

      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 84.2%

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

   3. Applied rewrites 84.2%

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

   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 70.5%

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

   6. Applied rewrites 70.5%

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

   8. Applied rewrites 63.4%

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

Alternative 11: 64.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \left|J\right| \cdot -2\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2}}}\right)\right)\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-137}:\\ \;\;\;\;\left(\sqrt{\frac{\left|U\right|}{\left(4 \cdot \left|J\right|\right) \cdot \left|J\right|} \cdot \left|U\right| - -1} \cdot t\_0\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -0.5 K)))
       (t_1 (* (fabs J) -2.0))
       (t_2 (cos (/ K 2.0)))
       (t_3
        (*
         (* (* -2.0 (fabs J)) t_2)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_3 (- INFINITY))
     (*
      -2.0
      (* (fabs J) (* (fabs U) (sqrt (/ 0.25 (pow (fabs J) 2.0))))))
     (if (<= t_3 -2e-137)
       (*
        (*
         (sqrt
          (-
           (* (/ (fabs U) (* (* 4.0 (fabs J)) (fabs J))) (fabs U))
           -1.0))
         t_0)
        t_1)
       (* t_0 t_1))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = fabs(J) * -2.0;
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * fabs(J)) * t_2) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -2.0 * (fabs(J) * (fabs(U) * sqrt((0.25 / pow(fabs(J), 2.0)))));
	} else if (t_3 <= -2e-137) {
		tmp = (sqrt((((fabs(U) / ((4.0 * fabs(J)) * fabs(J))) * fabs(U)) - -1.0)) * t_0) * t_1;
	} else {
		tmp = t_0 * t_1;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.abs(J) * -2.0;
	double t_2 = Math.cos((K / 2.0));
	double t_3 = ((-2.0 * Math.abs(J)) * t_2) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (Math.abs(J) * (Math.abs(U) * Math.sqrt((0.25 / Math.pow(Math.abs(J), 2.0)))));
	} else if (t_3 <= -2e-137) {
		tmp = (Math.sqrt((((Math.abs(U) / ((4.0 * Math.abs(J)) * Math.abs(J))) * Math.abs(U)) - -1.0)) * t_0) * t_1;
	} else {
		tmp = t_0 * t_1;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.fabs(J) * -2.0
	t_2 = math.cos((K / 2.0))
	t_3 = ((-2.0 * math.fabs(J)) * t_2) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_2)), 2.0)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = -2.0 * (math.fabs(J) * (math.fabs(U) * math.sqrt((0.25 / math.pow(math.fabs(J), 2.0)))))
	elif t_3 <= -2e-137:
		tmp = (math.sqrt((((math.fabs(U) / ((4.0 * math.fabs(J)) * math.fabs(J))) * math.fabs(U)) - -1.0)) * t_0) * t_1
	else:
		tmp = t_0 * t_1
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(abs(J) * -2.0)
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * abs(J)) * t_2) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(abs(J) * Float64(abs(U) * sqrt(Float64(0.25 / (abs(J) ^ 2.0))))));
	elseif (t_3 <= -2e-137)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(abs(U) / Float64(Float64(4.0 * abs(J)) * abs(J))) * abs(U)) - -1.0)) * t_0) * t_1);
	else
		tmp = Float64(t_0 * t_1);
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K));
	t_1 = abs(J) * -2.0;
	t_2 = cos((K / 2.0));
	t_3 = ((-2.0 * abs(J)) * t_2) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_2)) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -2.0 * (abs(J) * (abs(U) * sqrt((0.25 / (abs(J) ^ 2.0)))));
	elseif (t_3 <= -2e-137)
		tmp = (sqrt((((abs(U) / ((4.0 * abs(J)) * abs(J))) * abs(U)) - -1.0)) * t_0) * t_1;
	else
		tmp = t_0 * t_1;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] * N[Sqrt[N[(0.25 / N[Power[N[Abs[J], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-137], N[(N[(N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(4.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$0 * t$95$1), $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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 13.3%

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

   7. Applied rewrites 13.3%

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

   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))))) < -2e-137

   1. Initial program 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 60.4%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 60.4%

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 55.5%

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

   if -2e-137 < (*.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 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in J around inf

      \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)} \cdot \left(J \cdot -2\right) \]
   7. 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 50.7%

         \[\leadsto \cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right) \]

   8. Applied rewrites 50.7%

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

Alternative 12: 58.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0))))
  (*
   (copysign 1.0 J)
   (if (<=
        (*
         (* (* -2.0 (fabs J)) t_0)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0))))
        (- INFINITY))
     (*
      -2.0
      (* (fabs J) (* (fabs U) (sqrt (/ 0.25 (pow (fabs J) 2.0))))))
     (* (cos (* -0.5 K)) (* (fabs J) -2.0))))))

C

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

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * Math.abs(J)) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_0)), 2.0)))) <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (Math.abs(J) * (Math.abs(U) * Math.sqrt((0.25 / Math.pow(Math.abs(J), 2.0)))));
	} else {
		tmp = Math.cos((-0.5 * K)) * (Math.abs(J) * -2.0);
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * math.fabs(J)) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_0)), 2.0)))) <= -math.inf:
		tmp = -2.0 * (math.fabs(J) * (math.fabs(U) * math.sqrt((0.25 / math.pow(math.fabs(J), 2.0)))))
	else:
		tmp = math.cos((-0.5 * K)) * (math.fabs(J) * -2.0)
	return math.copysign(1.0, J) * tmp

Julia

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

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * abs(J)) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_0)) ^ 2.0)))) <= -Inf)
		tmp = -2.0 * (abs(J) * (abs(U) * sqrt((0.25 / (abs(J) ^ 2.0)))));
	else
		tmp = cos((-0.5 * K)) * (abs(J) * -2.0);
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] * N[Sqrt[N[(0.25 / N[Power[N[Abs[J], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $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 72.2%

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

   2. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 13.3%

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

   7. Applied rewrites 13.3%

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

   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 72.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 60.4%

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in J around inf

      \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)} \cdot \left(J \cdot -2\right) \]
   7. 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 50.7%

         \[\leadsto \cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right) \]

   8. Applied rewrites 50.7%

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

Alternative 13: 50.7% accurate, 3.2× speedup

Math

\[\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right) \]

FPCore

(FPCore (J K U)
  :precision binary64
  (* (cos (* -0.5 K)) (* J -2.0)))

C

double code(double J, double K, double U) {
	return cos((-0.5 * K)) * (J * -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
    code = cos(((-0.5d0) * k)) * (j * (-2.0d0))
end function

Java

public static double code(double J, double K, double U) {
	return Math.cos((-0.5 * K)) * (J * -2.0);
}

Python

def code(J, K, U):
	return math.cos((-0.5 * K)) * (J * -2.0)

Julia

function code(J, K, U)
	return Float64(cos(Float64(-0.5 * K)) * Float64(J * -2.0))
end

MATLAB

function tmp = code(J, K, U)
	tmp = cos((-0.5 * K)) * (J * -2.0);
end

Wolfram

code[J_, K_, U_] := N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.2%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 60.4%

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

5. Applied rewrites 54.4%

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

6. Taylor expanded in J around inf

   \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)} \cdot \left(J \cdot -2\right) \]
7. 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 50.7%

      \[\leadsto \cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot -2\right) \]

8. Applied rewrites 50.7%

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

Alternative 14: 27.2% accurate, 7.2× speedup

Math

\[\left(\left(-2 \cdot J\right) \cdot \left(1 - \left(0.125 - 0.0026041666666666665 \cdot \left(K \cdot K\right)\right) \cdot \left(K \cdot K\right)\right)\right) \cdot \sqrt{1} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (*
 (*
  (* -2.0 J)
  (- 1.0 (* (- 0.125 (* 0.0026041666666666665 (* K K))) (* K K))))
 (sqrt 1.0)))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * (1.0 - ((0.125 - (0.0026041666666666665 * (K * K))) * (K * K)))) * sqrt(1.0);
}

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
    code = (((-2.0d0) * j) * (1.0d0 - ((0.125d0 - (0.0026041666666666665d0 * (k * k))) * (k * k)))) * sqrt(1.0d0)
end function

Java

public static double code(double J, double K, double U) {
	return ((-2.0 * J) * (1.0 - ((0.125 - (0.0026041666666666665 * (K * K))) * (K * K)))) * Math.sqrt(1.0);
}

Python

def code(J, K, U):
	return ((-2.0 * J) * (1.0 - ((0.125 - (0.0026041666666666665 * (K * K))) * (K * K)))) * math.sqrt(1.0)

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * Float64(1.0 - Float64(Float64(0.125 - Float64(0.0026041666666666665 * Float64(K * K))) * Float64(K * K)))) * sqrt(1.0))
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * (1.0 - ((0.125 - (0.0026041666666666665 * (K * K))) * (K * K)))) * sqrt(1.0);
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[(1.0 - N[(N[(0.125 - N[(0.0026041666666666665 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.2%

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

2. Taylor expanded in J around inf

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in K around 0

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{1} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + \color{blue}{{K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \color{blue}{\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\color{blue}{\frac{1}{384} \cdot {K}^{2}} - \frac{1}{8}\right)\right)\right) \cdot \sqrt{1} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \color{blue}{\frac{1}{8}}\right)\right)\right) \cdot \sqrt{1} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot \sqrt{1} \]

   6. lower-pow.f64 27.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(0.0026041666666666665 \cdot {K}^{2} - 0.125\right)\right)\right) \cdot \sqrt{1} \]

7. Applied rewrites 27.1%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(0.0026041666666666665 \cdot {K}^{2} - 0.125\right)\right)}\right) \cdot \sqrt{1} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + \color{blue}{{K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \color{blue}{\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left({K}^{2}\right)\right) \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left({K}^{2}\right)\right) \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) \cdot \color{blue}{\left(\mathsf{neg}\left({K}^{2}\right)\right)}\right)\right) \cdot \sqrt{1} \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\mathsf{neg}\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2}\right)\right)\right)\right) \cdot \sqrt{1} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\mathsf{neg}\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot \color{blue}{{K}^{2}}\right)\right) \cdot \sqrt{1} \]

   8. lift--.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\mathsf{neg}\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot {K}^{2}\right)\right) \cdot \sqrt{1} \]

   9. sub-negate-rev N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot {K}^{2}\right) \cdot {\color{blue}{K}}^{2}\right)\right) \cdot \sqrt{1} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot {K}^{2}\right) \cdot \color{blue}{{K}^{2}}\right)\right) \cdot \sqrt{1} \]

   11. lower--.f64 27.1%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(0.125 - 0.0026041666666666665 \cdot {K}^{2}\right) \cdot {\color{blue}{K}}^{2}\right)\right) \cdot \sqrt{1} \]

   12. lift-pow.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot {K}^{2}\right) \cdot {K}^{2}\right)\right) \cdot \sqrt{1} \]

   13. unpow2 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot \left(K \cdot K\right)\right) \cdot {K}^{2}\right)\right) \cdot \sqrt{1} \]

   14. lower-*.f64 27.1%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(0.125 - 0.0026041666666666665 \cdot \left(K \cdot K\right)\right) \cdot {K}^{2}\right)\right) \cdot \sqrt{1} \]

   15. lift-pow.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot \left(K \cdot K\right)\right) \cdot {K}^{\color{blue}{2}}\right)\right) \cdot \sqrt{1} \]

   16. unpow2 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot \left(K \cdot K\right)\right) \cdot \left(K \cdot \color{blue}{K}\right)\right)\right) \cdot \sqrt{1} \]

   17. lower-*.f64 27.1%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(0.125 - 0.0026041666666666665 \cdot \left(K \cdot K\right)\right) \cdot \left(K \cdot \color{blue}{K}\right)\right)\right) \cdot \sqrt{1} \]

9. Applied rewrites 27.1%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \color{blue}{\left(0.125 - 0.0026041666666666665 \cdot \left(K \cdot K\right)\right) \cdot \left(K \cdot K\right)}\right)\right) \cdot \sqrt{1} \]
10. Add Preprocessing

Alternative 15: 27.1% accurate, 9.6× speedup

Math

\[\left(\left(-2 \cdot J\right) \cdot \left(1 - 0.125 \cdot \left(K \cdot K\right)\right)\right) \cdot \sqrt{1} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (* (* (* -2.0 J) (- 1.0 (* 0.125 (* K K)))) (sqrt 1.0)))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * (1.0 - (0.125 * (K * K)))) * sqrt(1.0);
}

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
    code = (((-2.0d0) * j) * (1.0d0 - (0.125d0 * (k * k)))) * sqrt(1.0d0)
end function

Java

public static double code(double J, double K, double U) {
	return ((-2.0 * J) * (1.0 - (0.125 * (K * K)))) * Math.sqrt(1.0);
}

Python

def code(J, K, U):
	return ((-2.0 * J) * (1.0 - (0.125 * (K * K)))) * math.sqrt(1.0)

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * Float64(1.0 - Float64(0.125 * Float64(K * K)))) * sqrt(1.0))
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * (1.0 - (0.125 * (K * K)))) * sqrt(1.0);
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[(1.0 - N[(0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.2%

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

2. Taylor expanded in J around inf

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in K around 0

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{1} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + \color{blue}{{K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \color{blue}{\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\color{blue}{\frac{1}{384} \cdot {K}^{2}} - \frac{1}{8}\right)\right)\right) \cdot \sqrt{1} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \color{blue}{\frac{1}{8}}\right)\right)\right) \cdot \sqrt{1} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot \sqrt{1} \]

   6. lower-pow.f64 27.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(0.0026041666666666665 \cdot {K}^{2} - 0.125\right)\right)\right) \cdot \sqrt{1} \]

7. Applied rewrites 27.1%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(0.0026041666666666665 \cdot {K}^{2} - 0.125\right)\right)}\right) \cdot \sqrt{1} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + \color{blue}{{K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \color{blue}{\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left({K}^{2}\right)\right) \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   4. lower--.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left({K}^{2}\right)\right) \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) \cdot \color{blue}{\left(\mathsf{neg}\left({K}^{2}\right)\right)}\right)\right) \cdot \sqrt{1} \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\mathsf{neg}\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) \cdot {K}^{2}\right)\right)\right)\right) \cdot \sqrt{1} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\mathsf{neg}\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot \color{blue}{{K}^{2}}\right)\right) \cdot \sqrt{1} \]

   8. lift--.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\mathsf{neg}\left(\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot {K}^{2}\right)\right) \cdot \sqrt{1} \]

   9. sub-negate-rev N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot {K}^{2}\right) \cdot {\color{blue}{K}}^{2}\right)\right) \cdot \sqrt{1} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot {K}^{2}\right) \cdot \color{blue}{{K}^{2}}\right)\right) \cdot \sqrt{1} \]

   11. lower--.f64 27.1%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(0.125 - 0.0026041666666666665 \cdot {K}^{2}\right) \cdot {\color{blue}{K}}^{2}\right)\right) \cdot \sqrt{1} \]

   12. lift-pow.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot {K}^{2}\right) \cdot {K}^{2}\right)\right) \cdot \sqrt{1} \]

   13. unpow2 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot \left(K \cdot K\right)\right) \cdot {K}^{2}\right)\right) \cdot \sqrt{1} \]

   14. lower-*.f64 27.1%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(0.125 - 0.0026041666666666665 \cdot \left(K \cdot K\right)\right) \cdot {K}^{2}\right)\right) \cdot \sqrt{1} \]

   15. lift-pow.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot \left(K \cdot K\right)\right) \cdot {K}^{\color{blue}{2}}\right)\right) \cdot \sqrt{1} \]

   16. unpow2 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(\frac{1}{8} - \frac{1}{384} \cdot \left(K \cdot K\right)\right) \cdot \left(K \cdot \color{blue}{K}\right)\right)\right) \cdot \sqrt{1} \]

   17. lower-*.f64 27.1%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \left(0.125 - 0.0026041666666666665 \cdot \left(K \cdot K\right)\right) \cdot \left(K \cdot \color{blue}{K}\right)\right)\right) \cdot \sqrt{1} \]

9. Applied rewrites 27.1%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - \color{blue}{\left(0.125 - 0.0026041666666666665 \cdot \left(K \cdot K\right)\right) \cdot \left(K \cdot K\right)}\right)\right) \cdot \sqrt{1} \]

10. Taylor expanded in K around 0

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

12. Applied rewrites 27.2%

    \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 - 0.125 \cdot \left(\color{blue}{K} \cdot K\right)\right)\right) \cdot \sqrt{1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(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)))))