Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.1% → 99.7%

Time: 7.2s

Alternatives: 17

Speedup: 0.4×

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: 73.1% 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: 99.7% accurate, 0.3× speedup

Math

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

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))))
  (*
   (copysign 1.0 J)
   (if (<= t_1 (- INFINITY))
     (* -2.0 (* (fabs U) (* t_2 (sqrt (/ 0.25 (pow t_2 2.0))))))
     (if (<= t_1 2e+306)
       (*
        (* (* (cos (* -0.5 K)) (fabs J)) -2.0)
        (sqrt
         (-
          (pow
           (/ (fabs U) (* (cos (* K -0.5)) (+ (fabs J) (fabs J))))
           2.0)
          -1.0)))
       (* 2.0 (* 0.5 (fabs U))))))))

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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -2.0 * (fabs(U) * (t_2 * sqrt((0.25 / pow(t_2, 2.0)))));
	} else if (t_1 <= 2e+306) {
		tmp = ((cos((-0.5 * K)) * fabs(J)) * -2.0) * sqrt((pow((fabs(U) / (cos((K * -0.5)) * (fabs(J) + fabs(J)))), 2.0) - -1.0));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	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 tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (Math.abs(U) * (t_2 * Math.sqrt((0.25 / Math.pow(t_2, 2.0)))));
	} else if (t_1 <= 2e+306) {
		tmp = ((Math.cos((-0.5 * K)) * Math.abs(J)) * -2.0) * Math.sqrt((Math.pow((Math.abs(U) / (Math.cos((K * -0.5)) * (Math.abs(J) + Math.abs(J)))), 2.0) - -1.0));
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	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))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -2.0 * (math.fabs(U) * (t_2 * math.sqrt((0.25 / math.pow(t_2, 2.0)))))
	elif t_1 <= 2e+306:
		tmp = ((math.cos((-0.5 * K)) * math.fabs(J)) * -2.0) * math.sqrt((math.pow((math.fabs(U) / (math.cos((K * -0.5)) * (math.fabs(J) + math.fabs(J)))), 2.0) - -1.0))
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	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))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(abs(U) * Float64(t_2 * sqrt(Float64(0.25 / (t_2 ^ 2.0))))));
	elseif (t_1 <= 2e+306)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * abs(J)) * -2.0) * sqrt(Float64((Float64(abs(U) / Float64(cos(Float64(K * -0.5)) * Float64(abs(J) + abs(J)))) ^ 2.0) - -1.0)));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	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));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -2.0 * (abs(U) * (t_2 * sqrt((0.25 / (t_2 ^ 2.0)))));
	elseif (t_1 <= 2e+306)
		tmp = ((cos((-0.5 * K)) * abs(J)) * -2.0) * sqrt((((abs(U) / (cos((K * -0.5)) * (abs(J) + abs(J)))) ^ 2.0) - -1.0));
	else
		tmp = 2.0 * (0.5 * abs(U));
	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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], 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], If[LessEqual[t$95$1, 2e+306], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 25.7%

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

   7. Applied rewrites 25.7%

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

   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))))) < 2e306

   1. Initial program 73.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. cosh-def N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   5. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.1%

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. cos-neg-rev N/A

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

      17. lift-cos.f64 85.1%

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

   7. Applied rewrites 85.1%

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

   9. Applied rewrites 73.1%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -0.5 K)))
       (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 U) (* t_3 (sqrt (/ 0.25 (pow t_3 2.0))))))
     (if (<= t_2 2e+306)
       (*
        (*
         (sqrt
          (-
           (pow (/ (fabs U) (* (+ (fabs J) (fabs J)) t_0)) 2.0)
           -1.0))
         (* (fabs J) -2.0))
        t_0)
       (* 2.0 (* 0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	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(U) * (t_3 * sqrt((0.25 / pow(t_3, 2.0)))));
	} else if (t_2 <= 2e+306) {
		tmp = (sqrt((pow((fabs(U) / ((fabs(J) + fabs(J)) * t_0)), 2.0) - -1.0)) * (fabs(J) * -2.0)) * t_0;
	} else {
		tmp = 2.0 * (0.5 * 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.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(U) * (t_3 * Math.sqrt((0.25 / Math.pow(t_3, 2.0)))));
	} else if (t_2 <= 2e+306) {
		tmp = (Math.sqrt((Math.pow((Math.abs(U) / ((Math.abs(J) + Math.abs(J)) * t_0)), 2.0) - -1.0)) * (Math.abs(J) * -2.0)) * t_0;
	} else {
		tmp = 2.0 * (0.5 * 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.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(U) * (t_3 * math.sqrt((0.25 / math.pow(t_3, 2.0)))))
	elif t_2 <= 2e+306:
		tmp = (math.sqrt((math.pow((math.fabs(U) / ((math.fabs(J) + math.fabs(J)) * t_0)), 2.0) - -1.0)) * (math.fabs(J) * -2.0)) * t_0
	else:
		tmp = 2.0 * (0.5 * 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 = 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(U) * Float64(t_3 * sqrt(Float64(0.25 / (t_3 ^ 2.0))))));
	elseif (t_2 <= 2e+306)
		tmp = Float64(Float64(sqrt(Float64((Float64(abs(U) / Float64(Float64(abs(J) + abs(J)) * t_0)) ^ 2.0) - -1.0)) * Float64(abs(J) * -2.0)) * t_0);
	else
		tmp = Float64(2.0 * Float64(0.5 * 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 = 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(U) * (t_3 * sqrt((0.25 / (t_3 ^ 2.0)))));
	elseif (t_2 <= 2e+306)
		tmp = (sqrt((((abs(U) / ((abs(J) + abs(J)) * t_0)) ^ 2.0) - -1.0)) * (abs(J) * -2.0)) * t_0;
	else
		tmp = 2.0 * (0.5 * 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[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[U], $MachinePrecision] * N[(t$95$3 * N[Sqrt[N[(0.25 / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[(N[(N[Sqrt[N[(N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 25.7%

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

   7. Applied rewrites 25.7%

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

   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))))) < 2e306

   1. Initial program 73.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. cosh-def N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 85.1%

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

   5. Applied rewrites 73.1%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* 0.5 K)))
       (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 (+ (fabs J) (fabs J))))
  (*
   (copysign 1.0 J)
   (if (<= t_2 (- INFINITY))
     (* -2.0 (* (fabs U) (* t_0 (sqrt (/ 0.25 (pow t_0 2.0))))))
     (if (<= t_2 2e+306)
       (*
        (*
         (sqrt
          (fma
           (/ (fabs U) t_3)
           (/ (fabs U) (* (fma (cos K) 0.5 0.5) t_3))
           1.0))
         (* (fabs J) -2.0))
        (cos (* -0.5 K)))
       (* 2.0 (* 0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((0.5 * K));
	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 = fabs(J) + fabs(J);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (fabs(U) * (t_0 * sqrt((0.25 / pow(t_0, 2.0)))));
	} else if (t_2 <= 2e+306) {
		tmp = (sqrt(fma((fabs(U) / t_3), (fabs(U) / (fma(cos(K), 0.5, 0.5) * t_3)), 1.0)) * (fabs(J) * -2.0)) * cos((-0.5 * K));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(0.5 * K))
	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 = Float64(abs(J) + abs(J))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(abs(U) * Float64(t_0 * sqrt(Float64(0.25 / (t_0 ^ 2.0))))));
	elseif (t_2 <= 2e+306)
		tmp = Float64(Float64(sqrt(fma(Float64(abs(U) / t_3), Float64(abs(U) / Float64(fma(cos(K), 0.5, 0.5) * t_3)), 1.0)) * Float64(abs(J) * -2.0)) * cos(Float64(-0.5 * K)));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(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[(N[Abs[J], $MachinePrecision] + N[Abs[J], $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[U], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(0.25 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[(N[(N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 25.7%

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

   7. Applied rewrites 25.7%

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

   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))))) < 2e306

   1. Initial program 73.1%

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

   3. Applied rewrites 61.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lift-/.f64 N/A

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

   5. Applied rewrites 73.0%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot \left|J\right|\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \cos \left(0.5 \cdot K\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(\left|U\right| \cdot \left(t\_3 \cdot \sqrt{\frac{0.25}{{t\_3}^{2}}}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-100}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(0.5 \cdot \frac{\left|U\right|}{\left|J\right|}\right)}^{2}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{\left|U\right|}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(4 \cdot \left(\left|J\right| \cdot \left|J\right|\right)\right)}, \left|U\right|, 1\right)} \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \left|J\right|\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0)))
       (t_1 (* (* -2.0 (fabs J)) t_0))
       (t_2
        (*
         t_1
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0)))))
       (t_3 (cos (* 0.5 K))))
  (*
   (copysign 1.0 J)
   (if (<= t_2 (- INFINITY))
     (* -2.0 (* (fabs U) (* t_3 (sqrt (/ 0.25 (pow t_3 2.0))))))
     (if (<= t_2 5e-100)
       (* t_1 (sqrt (+ 1.0 (pow (* 0.5 (/ (fabs U) (fabs J))) 2.0))))
       (if (<= t_2 2e+306)
         (*
          (*
           (sqrt
            (fma
             (/
              (fabs U)
              (* (fma (cos K) 0.5 0.5) (* 4.0 (* (fabs J) (fabs J)))))
             (fabs U)
             1.0))
           (* (cos (* -0.5 K)) (fabs J)))
          -2.0)
         (* 2.0 (* 0.5 (fabs U)))))))))

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;
	double t_2 = t_1 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double t_3 = cos((0.5 * K));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (fabs(U) * (t_3 * sqrt((0.25 / pow(t_3, 2.0)))));
	} else if (t_2 <= 5e-100) {
		tmp = t_1 * sqrt((1.0 + pow((0.5 * (fabs(U) / fabs(J))), 2.0)));
	} else if (t_2 <= 2e+306) {
		tmp = (sqrt(fma((fabs(U) / (fma(cos(K), 0.5, 0.5) * (4.0 * (fabs(J) * fabs(J))))), fabs(U), 1.0)) * (cos((-0.5 * K)) * fabs(J))) * -2.0;
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * abs(J)) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0))))
	t_3 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(abs(U) * Float64(t_3 * sqrt(Float64(0.25 / (t_3 ^ 2.0))))));
	elseif (t_2 <= 5e-100)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(abs(U) / abs(J))) ^ 2.0))));
	elseif (t_2 <= 2e+306)
		tmp = Float64(Float64(sqrt(fma(Float64(abs(U) / Float64(fma(cos(K), 0.5, 0.5) * Float64(4.0 * Float64(abs(J) * abs(J))))), abs(U), 1.0)) * Float64(cos(Float64(-0.5 * K)) * abs(J))) * -2.0);
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * 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[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 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$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[U], $MachinePrecision] * N[(t$95$3 * N[Sqrt[N[(0.25 / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-100], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[(N[(N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(4.0 * N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 25.7%

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

   7. Applied rewrites 25.7%

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

   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))))) < 5.0000000000000001e-100

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.1%

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

   4. Applied rewrites 64.1%

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

   if 5.0000000000000001e-100 < (*.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))))) < 2e306

   1. Initial program 73.1%

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

   3. Applied rewrites 61.4%

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

   5. Applied rewrites 61.4%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot \left|J\right|\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-1 \cdot \frac{\left|U\right| \cdot t\_0}{\left|t\_0\right|}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-100}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + {\left(0.5 \cdot \frac{\left|U\right|}{\left|J\right|}\right)}^{2}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{\left|U\right|}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(4 \cdot \left(\left|J\right| \cdot \left|J\right|\right)\right)}, \left|U\right|, 1\right)} \cdot \left(t\_0 \cdot \left|J\right|\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -0.5 K)))
       (t_1 (cos (/ K 2.0)))
       (t_2 (* (* -2.0 (fabs J)) t_1))
       (t_3
        (*
         t_2
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_3 (- INFINITY))
     (* -1.0 (/ (* (fabs U) t_0) (fabs t_0)))
     (if (<= t_3 5e-100)
       (* t_2 (sqrt (+ 1.0 (pow (* 0.5 (/ (fabs U) (fabs J))) 2.0))))
       (if (<= t_3 2e+306)
         (*
          (*
           (sqrt
            (fma
             (/
              (fabs U)
              (* (fma (cos K) 0.5 0.5) (* 4.0 (* (fabs J) (fabs J)))))
             (fabs U)
             1.0))
           (* t_0 (fabs J)))
          -2.0)
         (* 2.0 (* 0.5 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * fabs(J)) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -1.0 * ((fabs(U) * t_0) / fabs(t_0));
	} else if (t_3 <= 5e-100) {
		tmp = t_2 * sqrt((1.0 + pow((0.5 * (fabs(U) / fabs(J))), 2.0)));
	} else if (t_3 <= 2e+306) {
		tmp = (sqrt(fma((fabs(U) / (fma(cos(K), 0.5, 0.5) * (4.0 * (fabs(J) * fabs(J))))), fabs(U), 1.0)) * (t_0 * fabs(J))) * -2.0;
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * abs(J)) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-1.0 * Float64(Float64(abs(U) * t_0) / abs(t_0)));
	elseif (t_3 <= 5e-100)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(abs(U) / abs(J))) ^ 2.0))));
	elseif (t_3 <= 2e+306)
		tmp = Float64(Float64(sqrt(fma(Float64(abs(U) / Float64(fma(cos(K), 0.5, 0.5) * Float64(4.0 * Float64(abs(J) * abs(J))))), abs(U), 1.0)) * Float64(t_0 * abs(J))) * -2.0);
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(-1.0 * N[(N[(N[Abs[U], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e-100], N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+306], N[(N[(N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(4.0 * N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   6. Applied rewrites 12.5%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 25.7%

         \[\leadsto -1 \cdot \frac{U \cdot \cos \left(-0.5 \cdot K\right)}{\left|\cos \left(-0.5 \cdot K\right)\right|} \]

   9. Applied rewrites 25.7%

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

   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))))) < 5.0000000000000001e-100

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.1%

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

   4. Applied rewrites 64.1%

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

   if 5.0000000000000001e-100 < (*.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))))) < 2e306

   1. Initial program 73.1%

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

   3. Applied rewrites 61.4%

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

   5. Applied rewrites 61.4%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot \left|J\right|\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-1 \cdot \frac{\left|U\right| \cdot t\_0}{\left|t\_0\right|}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-100}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + {\left(0.5 \cdot \frac{\left|U\right|}{\left|J\right|}\right)}^{2}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{\left|U\right|}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(4 \cdot \left(\left|J\right| \cdot \left|J\right|\right)\right)}, \left|U\right|, 1\right)} \cdot t\_0\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -0.5 K)))
       (t_1 (cos (/ K 2.0)))
       (t_2 (* (* -2.0 (fabs J)) t_1))
       (t_3
        (*
         t_2
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_3 (- INFINITY))
     (* -1.0 (/ (* (fabs U) t_0) (fabs t_0)))
     (if (<= t_3 5e-100)
       (* t_2 (sqrt (+ 1.0 (pow (* 0.5 (/ (fabs U) (fabs J))) 2.0))))
       (if (<= t_3 2e+306)
         (*
          (*
           (sqrt
            (fma
             (/
              (fabs U)
              (* (fma (cos K) 0.5 0.5) (* 4.0 (* (fabs J) (fabs J)))))
             (fabs U)
             1.0))
           t_0)
          (* (fabs J) -2.0))
         (* 2.0 (* 0.5 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * fabs(J)) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -1.0 * ((fabs(U) * t_0) / fabs(t_0));
	} else if (t_3 <= 5e-100) {
		tmp = t_2 * sqrt((1.0 + pow((0.5 * (fabs(U) / fabs(J))), 2.0)));
	} else if (t_3 <= 2e+306) {
		tmp = (sqrt(fma((fabs(U) / (fma(cos(K), 0.5, 0.5) * (4.0 * (fabs(J) * fabs(J))))), fabs(U), 1.0)) * t_0) * (fabs(J) * -2.0);
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * abs(J)) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-1.0 * Float64(Float64(abs(U) * t_0) / abs(t_0)));
	elseif (t_3 <= 5e-100)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(abs(U) / abs(J))) ^ 2.0))));
	elseif (t_3 <= 2e+306)
		tmp = Float64(Float64(sqrt(fma(Float64(abs(U) / Float64(fma(cos(K), 0.5, 0.5) * Float64(4.0 * Float64(abs(J) * abs(J))))), abs(U), 1.0)) * t_0) * Float64(abs(J) * -2.0));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(-1.0 * N[(N[(N[Abs[U], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e-100], N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+306], N[(N[(N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(4.0 * N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   6. Applied rewrites 12.5%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 25.7%

         \[\leadsto -1 \cdot \frac{U \cdot \cos \left(-0.5 \cdot K\right)}{\left|\cos \left(-0.5 \cdot K\right)\right|} \]

   9. Applied rewrites 25.7%

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

   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))))) < 5.0000000000000001e-100

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.1%

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

   4. Applied rewrites 64.1%

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

   if 5.0000000000000001e-100 < (*.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))))) < 2e306

   1. Initial program 73.1%

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

   3. Applied rewrites 61.4%

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

   5. Applied rewrites 61.3%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -0.5 K)))
       (t_1 (cos (/ K 2.0)))
       (t_2 (* (* -2.0 (fabs J)) t_1))
       (t_3
        (*
         t_2
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_3 (- INFINITY))
     (* -1.0 (/ (* (fabs U) t_0) (fabs t_0)))
     (if (<= t_3 2e+306)
       (* t_2 (sqrt (+ 1.0 (pow (* 0.5 (/ (fabs U) (fabs J))) 2.0))))
       (* 2.0 (* 0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * fabs(J)) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -1.0 * ((fabs(U) * t_0) / fabs(t_0));
	} else if (t_3 <= 2e+306) {
		tmp = t_2 * sqrt((1.0 + pow((0.5 * (fabs(U) / fabs(J))), 2.0)));
	} else {
		tmp = 2.0 * (0.5 * 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.cos((K / 2.0));
	double t_2 = (-2.0 * Math.abs(J)) * t_1;
	double t_3 = t_2 * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -1.0 * ((Math.abs(U) * t_0) / Math.abs(t_0));
	} else if (t_3 <= 2e+306) {
		tmp = t_2 * Math.sqrt((1.0 + Math.pow((0.5 * (Math.abs(U) / Math.abs(J))), 2.0)));
	} else {
		tmp = 2.0 * (0.5 * 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.cos((K / 2.0))
	t_2 = (-2.0 * math.fabs(J)) * t_1
	t_3 = t_2 * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = -1.0 * ((math.fabs(U) * t_0) / math.fabs(t_0))
	elif t_3 <= 2e+306:
		tmp = t_2 * math.sqrt((1.0 + math.pow((0.5 * (math.fabs(U) / math.fabs(J))), 2.0)))
	else:
		tmp = 2.0 * (0.5 * 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 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * abs(J)) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-1.0 * Float64(Float64(abs(U) * t_0) / abs(t_0)));
	elseif (t_3 <= 2e+306)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(abs(U) / abs(J))) ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(0.5 * 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 = cos((K / 2.0));
	t_2 = (-2.0 * abs(J)) * t_1;
	t_3 = t_2 * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -1.0 * ((abs(U) * t_0) / abs(t_0));
	elseif (t_3 <= 2e+306)
		tmp = t_2 * sqrt((1.0 + ((0.5 * (abs(U) / abs(J))) ^ 2.0)));
	else
		tmp = 2.0 * (0.5 * 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(-1.0 * N[(N[(N[Abs[U], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+306], N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   6. Applied rewrites 12.5%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 25.7%

         \[\leadsto -1 \cdot \frac{U \cdot \cos \left(-0.5 \cdot K\right)}{\left|\cos \left(-0.5 \cdot K\right)\right|} \]

   9. Applied rewrites 25.7%

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

   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))))) < 2e306

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.1%

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

   4. Applied rewrites 64.1%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \frac{\left|U\right|}{J + J}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(J + J\right) \cdot \left(\frac{\left|U\right|}{J} \cdot -0.5\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{t\_3}{\left(0.5 - -0.5\right) \cdot \left(J + J\right)}, \left|U\right|, 1\right)} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(-0.5 \cdot K\right)\\ \mathbf{elif}\;t\_2 \leq -4000:\\ \;\;\;\;-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{\left(\left|U\right|\right)}^{2}}{{J}^{2}}}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + \frac{t\_3 \cdot \left|U\right|}{\left(0.5 + 0.5\right) \cdot \left(J + J\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0)))
       (t_1 (* (* -2.0 J) t_0))
       (t_2
        (*
         t_1
         (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 J) t_0)) 2.0)))))
       (t_3 (/ (fabs U) (+ J J))))
  (if (<= t_2 (- INFINITY))
    (* (+ J J) (* (/ (fabs U) J) -0.5))
    (if (<= t_2 -2e+29)
      (*
       (*
        (sqrt (fma (/ t_3 (* (- 0.5 -0.5) (+ J J))) (fabs U) 1.0))
        (* J -2.0))
       (cos (* -0.5 K)))
      (if (<= t_2 -4000.0)
        (*
         -2.0
         (*
          J
          (sqrt (+ 1.0 (* 0.25 (/ (pow (fabs U) 2.0) (pow J 2.0)))))))
        (if (<= t_2 2e+306)
          (*
           t_1
           (sqrt
            (+ 1.0 (/ (* t_3 (fabs U)) (* (+ 0.5 0.5) (+ J J))))))
          (* 2.0 (* 0.5 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_0)), 2.0)));
	double t_3 = fabs(U) / (J + J);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (J + J) * ((fabs(U) / J) * -0.5);
	} else if (t_2 <= -2e+29) {
		tmp = (sqrt(fma((t_3 / ((0.5 - -0.5) * (J + J))), fabs(U), 1.0)) * (J * -2.0)) * cos((-0.5 * K));
	} else if (t_2 <= -4000.0) {
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * (pow(fabs(U), 2.0) / pow(J, 2.0))))));
	} else if (t_2 <= 2e+306) {
		tmp = t_1 * sqrt((1.0 + ((t_3 * fabs(U)) / ((0.5 + 0.5) * (J + J)))));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_3 = Float64(abs(U) / Float64(J + J))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(J + J) * Float64(Float64(abs(U) / J) * -0.5));
	elseif (t_2 <= -2e+29)
		tmp = Float64(Float64(sqrt(fma(Float64(t_3 / Float64(Float64(0.5 - -0.5) * Float64(J + J))), abs(U), 1.0)) * Float64(J * -2.0)) * cos(Float64(-0.5 * K)));
	elseif (t_2 <= -4000.0)
		tmp = Float64(-2.0 * Float64(J * sqrt(Float64(1.0 + Float64(0.25 * Float64((abs(U) ^ 2.0) / (J ^ 2.0)))))));
	elseif (t_2 <= 2e+306)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + Float64(Float64(t_3 * abs(U)) / Float64(Float64(0.5 + 0.5) * Float64(J + J))))));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return 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[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[U], $MachinePrecision] / N[(J + J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(J + J), $MachinePrecision] * N[(N[(N[Abs[U], $MachinePrecision] / J), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+29], N[(N[(N[Sqrt[N[(N[(t$95$3 / N[(N[(0.5 - -0.5), $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4000.0], N[(-2.0 * N[(J * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[N[Abs[U], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[J, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[(t$95$1 * N[Sqrt[N[(1.0 + N[(N[(t$95$3 * N[Abs[U], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 + 0.5), $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * 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 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 19.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. count-2-rev N/A

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

       5. lift-+.f64 N/A

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

       6. lower-*.f64 19.9%

          \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J}\right)} \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\frac{U}{J} \cdot -0.5\right)} \]

   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))))) < -1.9999999999999998e29

   1. Initial program 73.1%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 56.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 62.4%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 62.4%

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

   8. Applied rewrites 62.4%

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

   if -1.9999999999999998e29 < (*.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))))) < -4e3

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 33.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

   10. Applied rewrites 33.7%

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

   if -4e3 < (*.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))))) < 2e306

   1. Initial program 73.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   3. Applied rewrites 70.2%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 62.5%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 83.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0)))
       (t_1 (* (* -2.0 J) t_0))
       (t_2
        (*
         t_1
         (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 J) t_0)) 2.0)))))
       (t_3 (/ (fabs U) J)))
  (if (<= t_2 (- INFINITY))
    (* (+ J J) (* t_3 -0.5))
    (if (<= t_2 2e+306)
      (* t_1 (sqrt (+ 1.0 (pow (* 0.5 t_3) 2.0))))
      (* 2.0 (* 0.5 (fabs U)))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_0)), 2.0)));
	double t_3 = fabs(U) / J;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (J + J) * (t_3 * -0.5);
	} else if (t_2 <= 2e+306) {
		tmp = t_1 * sqrt((1.0 + pow((0.5 * t_3), 2.0)));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return 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 * J) * t_0;
	double t_2 = t_1 * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * J) * t_0)), 2.0)));
	double t_3 = Math.abs(U) / J;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (J + J) * (t_3 * -0.5);
	} else if (t_2 <= 2e+306) {
		tmp = t_1 * Math.sqrt((1.0 + Math.pow((0.5 * t_3), 2.0)));
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = (-2.0 * J) * t_0
	t_2 = t_1 * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * J) * t_0)), 2.0)))
	t_3 = math.fabs(U) / J
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (J + J) * (t_3 * -0.5)
	elif t_2 <= 2e+306:
		tmp = t_1 * math.sqrt((1.0 + math.pow((0.5 * t_3), 2.0)))
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_3 = Float64(abs(U) / J)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(J + J) * Float64(t_3 * -0.5));
	elseif (t_2 <= 2e+306)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(0.5 * t_3) ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = (-2.0 * J) * t_0;
	t_2 = t_1 * sqrt((1.0 + ((abs(U) / ((2.0 * J) * t_0)) ^ 2.0)));
	t_3 = abs(U) / J;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (J + J) * (t_3 * -0.5);
	elseif (t_2 <= 2e+306)
		tmp = t_1 * sqrt((1.0 + ((0.5 * t_3) ^ 2.0)));
	else
		tmp = 2.0 * (0.5 * abs(U));
	end
	tmp_2 = 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[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[U], $MachinePrecision] / J), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(J + J), $MachinePrecision] * N[(t$95$3 * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * 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 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 19.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. count-2-rev N/A

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

       5. lift-+.f64 N/A

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

       6. lower-*.f64 19.9%

          \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J}\right)} \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\frac{U}{J} \cdot -0.5\right)} \]

   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))))) < 2e306

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.1%

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

   4. Applied rewrites 64.1%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0)))
       (t_1
        (*
         (* (* -2.0 J) t_0)
         (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 J) t_0)) 2.0))))))
  (if (<= t_1 (- INFINITY))
    (* (+ J J) (* (/ (fabs U) J) -0.5))
    (if (<= t_1 2e+306)
      (*
       (*
        (sqrt
         (fma
          (/ (/ (fabs U) (+ J J)) (* (- 0.5 -0.5) (+ J J)))
          (fabs U)
          1.0))
        (* J -2.0))
       (cos (* -0.5 K)))
      (* 2.0 (* 0.5 (fabs U)))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (J + J) * ((fabs(U) / J) * -0.5);
	} else if (t_1 <= 2e+306) {
		tmp = (sqrt(fma(((fabs(U) / (J + J)) / ((0.5 - -0.5) * (J + J))), fabs(U), 1.0)) * (J * -2.0)) * cos((-0.5 * K));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(J + J) * Float64(Float64(abs(U) / J) * -0.5));
	elseif (t_1 <= 2e+306)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(abs(U) / Float64(J + J)) / Float64(Float64(0.5 - -0.5) * Float64(J + J))), abs(U), 1.0)) * Float64(J * -2.0)) * cos(Float64(-0.5 * K)));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return 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 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(J + J), $MachinePrecision] * N[(N[(N[Abs[U], $MachinePrecision] / J), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(N[(N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(J + J), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - -0.5), $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * 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 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 19.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. count-2-rev N/A

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

       5. lift-+.f64 N/A

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

       6. lower-*.f64 19.9%

          \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J}\right)} \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\frac{U}{J} \cdot -0.5\right)} \]

   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))))) < 2e306

   1. Initial program 73.1%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 56.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 62.4%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 62.4%

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

   8. Applied rewrites 62.4%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(J + J\right) \cdot \left(\frac{\left|U\right|}{J} \cdot -0.5\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-133}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left(\frac{\left|U\right|}{\left(0.5 - -0.5\right) \cdot \left(4 \cdot \left(J \cdot J\right)\right)}, \left|U\right|, 1\right)} \cdot J\right) \cdot -2\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(t\_2 \cdot J\right) \cdot -2\right) \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0)))
       (t_1
        (*
         (* (* -2.0 J) t_0)
         (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 J) t_0)) 2.0)))))
       (t_2 (cos (* -0.5 K))))
  (if (<= t_1 (- INFINITY))
    (* (+ J J) (* (/ (fabs U) J) -0.5))
    (if (<= t_1 -2e-133)
      (*
       (*
        (*
         (sqrt
          (fma
           (/ (fabs U) (* (- 0.5 -0.5) (* 4.0 (* J J))))
           (fabs U)
           1.0))
         J)
        -2.0)
       t_2)
      (if (<= t_1 2e+306)
        (* (* (* t_2 J) -2.0) (sqrt 1.0))
        (* 2.0 (* 0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_0)), 2.0)));
	double t_2 = cos((-0.5 * K));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (J + J) * ((fabs(U) / J) * -0.5);
	} else if (t_1 <= -2e-133) {
		tmp = ((sqrt(fma((fabs(U) / ((0.5 - -0.5) * (4.0 * (J * J)))), fabs(U), 1.0)) * J) * -2.0) * t_2;
	} else if (t_1 <= 2e+306) {
		tmp = ((t_2 * J) * -2.0) * sqrt(1.0);
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(J + J) * Float64(Float64(abs(U) / J) * -0.5));
	elseif (t_1 <= -2e-133)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(abs(U) / Float64(Float64(0.5 - -0.5) * Float64(4.0 * Float64(J * J)))), abs(U), 1.0)) * J) * -2.0) * t_2);
	elseif (t_1 <= 2e+306)
		tmp = Float64(Float64(Float64(t_2 * J) * -2.0) * sqrt(1.0));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return 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 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(J + J), $MachinePrecision] * N[(N[(N[Abs[U], $MachinePrecision] / J), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-133], N[(N[(N[(N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(0.5 - -0.5), $MachinePrecision] * N[(4.0 * N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(N[(N[(t$95$2 * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 19.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. count-2-rev N/A

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

       5. lift-+.f64 N/A

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

       6. lower-*.f64 19.9%

          \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J}\right)} \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\frac{U}{J} \cdot -0.5\right)} \]

   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))))) < -2.0000000000000001e-133

   1. Initial program 73.1%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 56.3%

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

   if -2.0000000000000001e-133 < (*.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))))) < 2e306

   1. Initial program 73.1%

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

   2. 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 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.3%

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lift-*.f64 52.3%

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

   6. Applied rewrites 52.3%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{1}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(J + J\right) \cdot \left(\frac{\left|U\right|}{J} \cdot -0.5\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-133}:\\ \;\;\;\;-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{\left(\left|U\right|\right)}^{2}}{{J}^{2}}}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (* (* (* (cos (* -0.5 K)) J) -2.0) (sqrt 1.0)))
       (t_1 (cos (/ K 2.0)))
       (t_2
        (*
         (* (* -2.0 J) t_1)
         (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 J) t_1)) 2.0))))))
  (if (<= t_2 (- INFINITY))
    (* (+ J J) (* (/ (fabs U) J) -0.5))
    (if (<= t_2 -5e+125)
      t_0
      (if (<= t_2 -2e-133)
        (*
         -2.0
         (*
          J
          (sqrt (+ 1.0 (* 0.25 (/ (pow (fabs U) 2.0) (pow J 2.0)))))))
        (if (<= t_2 2e+306) t_0 (* 2.0 (* 0.5 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = ((cos((-0.5 * K)) * J) * -2.0) * sqrt(1.0);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (J + J) * ((fabs(U) / J) * -0.5);
	} else if (t_2 <= -5e+125) {
		tmp = t_0;
	} else if (t_2 <= -2e-133) {
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * (pow(fabs(U), 2.0) / pow(J, 2.0))))));
	} else if (t_2 <= 2e+306) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = ((Math.cos((-0.5 * K)) * J) * -2.0) * Math.sqrt(1.0);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (J + J) * ((Math.abs(U) / J) * -0.5);
	} else if (t_2 <= -5e+125) {
		tmp = t_0;
	} else if (t_2 <= -2e-133) {
		tmp = -2.0 * (J * Math.sqrt((1.0 + (0.25 * (Math.pow(Math.abs(U), 2.0) / Math.pow(J, 2.0))))));
	} else if (t_2 <= 2e+306) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = ((math.cos((-0.5 * K)) * J) * -2.0) * math.sqrt(1.0)
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J) * t_1) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * J) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (J + J) * ((math.fabs(U) / J) * -0.5)
	elif t_2 <= -5e+125:
		tmp = t_0
	elif t_2 <= -2e-133:
		tmp = -2.0 * (J * math.sqrt((1.0 + (0.25 * (math.pow(math.fabs(U), 2.0) / math.pow(J, 2.0))))))
	elif t_2 <= 2e+306:
		tmp = t_0
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J) * -2.0) * sqrt(1.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(J + J) * Float64(Float64(abs(U) / J) * -0.5));
	elseif (t_2 <= -5e+125)
		tmp = t_0;
	elseif (t_2 <= -2e-133)
		tmp = Float64(-2.0 * Float64(J * sqrt(Float64(1.0 + Float64(0.25 * Float64((abs(U) ^ 2.0) / (J ^ 2.0)))))));
	elseif (t_2 <= 2e+306)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = ((cos((-0.5 * K)) * J) * -2.0) * sqrt(1.0);
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + ((abs(U) / ((2.0 * J) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (J + J) * ((abs(U) / J) * -0.5);
	elseif (t_2 <= -5e+125)
		tmp = t_0;
	elseif (t_2 <= -2e-133)
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * ((abs(U) ^ 2.0) / (J ^ 2.0))))));
	elseif (t_2 <= 2e+306)
		tmp = t_0;
	else
		tmp = 2.0 * (0.5 * abs(U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(J + J), $MachinePrecision] * N[(N[(N[Abs[U], $MachinePrecision] / J), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e+125], t$95$0, If[LessEqual[t$95$2, -2e-133], N[(-2.0 * N[(J * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[N[Abs[U], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[J, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+306], t$95$0, N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 19.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. count-2-rev N/A

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

       5. lift-+.f64 N/A

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

       6. lower-*.f64 19.9%

          \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J}\right)} \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\frac{U}{J} \cdot -0.5\right)} \]

   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))))) < -4.9999999999999996e125 or -2.0000000000000001e-133 < (*.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))))) < 2e306

   1. Initial program 73.1%

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

   2. 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 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.3%

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lift-*.f64 52.3%

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

   6. Applied rewrites 52.3%

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

   if -4.9999999999999996e125 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e-133

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 33.7%

         \[\leadsto -2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \]

   10. Applied rewrites 33.7%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 71.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(J + J\right) \cdot \left(\frac{\left|U\right|}{J} \cdot -0.5\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0)))
       (t_1
        (*
         (* (* -2.0 J) t_0)
         (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 J) t_0)) 2.0))))))
  (if (<= t_1 (- INFINITY))
    (* (+ J J) (* (/ (fabs U) J) -0.5))
    (if (<= t_1 2e+306)
      (* (* (* (cos (* -0.5 K)) J) -2.0) (sqrt 1.0))
      (* 2.0 (* 0.5 (fabs U)))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (J + J) * ((fabs(U) / J) * -0.5);
	} else if (t_1 <= 2e+306) {
		tmp = ((cos((-0.5 * K)) * J) * -2.0) * sqrt(1.0);
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return 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 * J) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (J + J) * ((Math.abs(U) / J) * -0.5);
	} else if (t_1 <= 2e+306) {
		tmp = ((Math.cos((-0.5 * K)) * J) * -2.0) * Math.sqrt(1.0);
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (J + J) * ((math.fabs(U) / J) * -0.5)
	elif t_1 <= 2e+306:
		tmp = ((math.cos((-0.5 * K)) * J) * -2.0) * math.sqrt(1.0)
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(J + J) * Float64(Float64(abs(U) / J) * -0.5));
	elseif (t_1 <= 2e+306)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J) * -2.0) * sqrt(1.0));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (J + J) * ((abs(U) / J) * -0.5);
	elseif (t_1 <= 2e+306)
		tmp = ((cos((-0.5 * K)) * J) * -2.0) * sqrt(1.0);
	else
		tmp = 2.0 * (0.5 * abs(U));
	end
	tmp_2 = 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 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(J + J), $MachinePrecision] * N[(N[(N[Abs[U], $MachinePrecision] / J), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * 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 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 19.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. count-2-rev N/A

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

       5. lift-+.f64 N/A

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

       6. lower-*.f64 19.9%

          \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J}\right)} \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\frac{U}{J} \cdot -0.5\right)} \]

   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))))) < 2e306

   1. Initial program 73.1%

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

   2. 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 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.3%

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lift-*.f64 52.3%

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

   6. Applied rewrites 52.3%

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

   if 2e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|J\right| + \left|J\right|\right) \cdot \left(\frac{\left|U\right|}{\left|J\right|} \cdot -0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\left|J\right|, -2, \left(\mathsf{fma}\left(\left(K \cdot K\right) \cdot \left|J\right|, -0.005208333333333333, 0.25 \cdot \left|J\right|\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (* (+ (fabs J) (fabs J)) (* (/ (fabs U) (fabs J)) -0.5)))
       (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))))))
  (*
   (copysign 1.0 J)
   (if (<= t_2 (- INFINITY))
     t_0
     (if (<= t_2 -5e+35)
       (*
        (fma
         (fabs J)
         -2.0
         (*
          (*
           (fma
            (* (* K K) (fabs J))
            -0.005208333333333333
            (* 0.25 (fabs J)))
           K)
          K))
        (sqrt 1.0))
       (if (<= t_2 -1e-202) t_0 (* 2.0 (* 0.5 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = (fabs(J) + fabs(J)) * ((fabs(U) / fabs(J)) * -0.5);
	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 tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= -5e+35) {
		tmp = fma(fabs(J), -2.0, ((fma(((K * K) * fabs(J)), -0.005208333333333333, (0.25 * fabs(J))) * K) * K)) * sqrt(1.0);
	} else if (t_2 <= -1e-202) {
		tmp = t_0;
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(Float64(abs(J) + abs(J)) * Float64(Float64(abs(U) / abs(J)) * -0.5))
	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))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= -5e+35)
		tmp = Float64(fma(abs(J), -2.0, Float64(Float64(fma(Float64(Float64(K * K) * abs(J)), -0.005208333333333333, Float64(0.25 * abs(J))) * K) * K)) * sqrt(1.0));
	elseif (t_2 <= -1e-202)
		tmp = t_0;
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, -5e+35], N[(N[(N[Abs[J], $MachinePrecision] * -2.0 + N[(N[(N[(N[(N[(K * K), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -0.005208333333333333 + N[(0.25 * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * K), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-202], t$95$0, N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $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 or -5.0000000000000002e35 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e-202

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 19.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. count-2-rev N/A

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

       5. lift-+.f64 N/A

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

       6. lower-*.f64 19.9%

          \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J}\right)} \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\frac{U}{J} \cdot -0.5\right)} \]

   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))))) < -5.0000000000000002e35

   1. Initial program 73.1%

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

   2. 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 52.3%

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

   5. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(-2 \cdot J + {K}^{2} \cdot \left(\frac{-1}{192} \cdot \left(J \cdot {K}^{2}\right) + \frac{1}{4} \cdot J\right)\right)} \cdot \sqrt{1} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{J}, {K}^{2} \cdot \left(\frac{-1}{192} \cdot \left(J \cdot {K}^{2}\right) + \frac{1}{4} \cdot J\right)\right) \cdot \sqrt{1} \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, {K}^{2} \cdot \left(\frac{-1}{192} \cdot \left(J \cdot {K}^{2}\right) + \frac{1}{4} \cdot J\right)\right) \cdot \sqrt{1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, {K}^{2} \cdot \left(\frac{-1}{192} \cdot \left(J \cdot {K}^{2}\right) + \frac{1}{4} \cdot J\right)\right) \cdot \sqrt{1} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, {K}^{2} \cdot \mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right)\right) \cdot \sqrt{1} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, {K}^{2} \cdot \mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right)\right) \cdot \sqrt{1} \]

      6. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, {K}^{2} \cdot \mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right)\right) \cdot \sqrt{1} \]

      7. lower-*.f64 28.5%

         \[\leadsto \mathsf{fma}\left(-2, J, {K}^{2} \cdot \mathsf{fma}\left(-0.005208333333333333, J \cdot {K}^{2}, 0.25 \cdot J\right)\right) \cdot \sqrt{1} \]

   7. Applied rewrites 28.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, J, {K}^{2} \cdot \mathsf{fma}\left(-0.005208333333333333, J \cdot {K}^{2}, 0.25 \cdot J\right)\right)} \cdot \sqrt{1} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot J + \color{blue}{{K}^{2} \cdot \mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right)}\right) \cdot \sqrt{1} \]

      2. *-commutative N/A

         \[\leadsto \left(J \cdot -2 + \color{blue}{{K}^{2}} \cdot \mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right)\right) \cdot \sqrt{1} \]

      3. lower-fma.f64 28.5%

         \[\leadsto \mathsf{fma}\left(J, \color{blue}{-2}, {K}^{2} \cdot \mathsf{fma}\left(-0.005208333333333333, J \cdot {K}^{2}, 0.25 \cdot J\right)\right) \cdot \sqrt{1} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, -2, {K}^{2} \cdot \mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right)\right) \cdot \sqrt{1} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(J, -2, \mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right) \cdot {K}^{2}\right) \cdot \sqrt{1} \]

      6. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, -2, \mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right) \cdot {K}^{2}\right) \cdot \sqrt{1} \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(J, -2, \mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right) \cdot \left(K \cdot K\right)\right) \cdot \sqrt{1} \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left(\frac{-1}{192}, J \cdot {K}^{2}, \frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      10. lower-*.f64 28.5%

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left(-0.005208333333333333, J \cdot {K}^{2}, 0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      11. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\left(\frac{-1}{192} \cdot \left(J \cdot {K}^{2}\right) + \frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\left(\left(J \cdot {K}^{2}\right) \cdot \frac{-1}{192} + \frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      13. lower-fma.f64 28.5%

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left(J \cdot {K}^{2}, -0.005208333333333333, 0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left(J \cdot {K}^{2}, \frac{-1}{192}, \frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left({K}^{2} \cdot J, \frac{-1}{192}, \frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      16. lower-*.f64 28.5%

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left({K}^{2} \cdot J, -0.005208333333333333, 0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      17. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left({K}^{2} \cdot J, \frac{-1}{192}, \frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left(\left(K \cdot K\right) \cdot J, \frac{-1}{192}, \frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

      19. lower-*.f64 28.5%

          \[\leadsto \mathsf{fma}\left(J, -2, \left(\mathsf{fma}\left(\left(K \cdot K\right) \cdot J, -0.005208333333333333, 0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

   9. Applied rewrites 28.5%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{-2}, \left(\mathsf{fma}\left(\left(K \cdot K\right) \cdot J, -0.005208333333333333, 0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot \sqrt{1} \]

   if -1e-202 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \left(\left|J\right| + \left|J\right|\right) \cdot \left(\frac{\left|U\right|}{\left|J\right|} \cdot -0.5\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -8 \cdot 10^{+307}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.25 \cdot \left|J\right|\right) \cdot K, K, t\_1\right) \cdot \sqrt{1}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0)))
       (t_1 (* -2.0 (fabs J)))
       (t_2
        (*
         (* t_1 t_0)
         (sqrt
          (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0)))))
       (t_3 (* (+ (fabs J) (fabs J)) (* (/ (fabs U) (fabs J)) -0.5))))
  (*
   (copysign 1.0 J)
   (if (<= t_2 -8e+307)
     t_3
     (if (<= t_2 -5e+35)
       (* (fma (* (* 0.25 (fabs J)) K) K t_1) (sqrt 1.0))
       (if (<= t_2 -1e-202) t_3 (* 2.0 (* 0.5 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = -2.0 * fabs(J);
	double t_2 = (t_1 * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double t_3 = (fabs(J) + fabs(J)) * ((fabs(U) / fabs(J)) * -0.5);
	double tmp;
	if (t_2 <= -8e+307) {
		tmp = t_3;
	} else if (t_2 <= -5e+35) {
		tmp = fma(((0.25 * fabs(J)) * K), K, t_1) * sqrt(1.0);
	} else if (t_2 <= -1e-202) {
		tmp = t_3;
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(-2.0 * abs(J))
	t_2 = Float64(Float64(t_1 * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0))))
	t_3 = Float64(Float64(abs(J) + abs(J)) * Float64(Float64(abs(U) / abs(J)) * -0.5))
	tmp = 0.0
	if (t_2 <= -8e+307)
		tmp = t_3;
	elseif (t_2 <= -5e+35)
		tmp = Float64(fma(Float64(Float64(0.25 * abs(J)) * K), K, t_1) * sqrt(1.0));
	elseif (t_2 <= -1e-202)
		tmp = t_3;
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * 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$3 = N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision] * -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$2, -8e+307], t$95$3, If[LessEqual[t$95$2, -5e+35], N[(N[(N[(N[(0.25 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * K), $MachinePrecision] * K + t$95$1), $MachinePrecision] * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-202], t$95$3, N[(2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $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))))) < -7.9999999999999999e307 or -5.0000000000000002e35 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e-202

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 19.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. count-2-rev N/A

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

       5. lift-+.f64 N/A

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

       6. lower-*.f64 19.9%

          \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J}\right)} \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\frac{U}{J} \cdot -0.5\right)} \]

   if -7.9999999999999999e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e35

   1. Initial program 73.1%

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

   2. 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 52.3%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 28.4%

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

   7. Applied rewrites 28.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 28.5%

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

   9. Applied rewrites 28.5%

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

   if -1e-202 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 45.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\left(J + J\right) \cdot \left(\frac{\left|U\right|}{J} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2.0))))
  (if (<=
       (*
        (* (* -2.0 J) t_0)
        (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 J) t_0)) 2.0))))
       -1e-202)
    (* (+ J J) (* (/ (fabs U) J) -0.5))
    (* 2.0 (* 0.5 (fabs U))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_0)), 2.0)))) <= -1e-202) {
		tmp = (J + J) * ((fabs(U) / J) * -0.5);
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((abs(u) / ((2.0d0 * j) * t_0)) ** 2.0d0)))) <= (-1d-202)) then
        tmp = (j + j) * ((abs(u) / j) * (-0.5d0))
    else
        tmp = 2.0d0 * (0.5d0 * abs(u))
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * J) * t_0)), 2.0)))) <= -1e-202) {
		tmp = (J + J) * ((Math.abs(U) / J) * -0.5);
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * J) * t_0)), 2.0)))) <= -1e-202:
		tmp = (J + J) * ((math.fabs(U) / J) * -0.5)
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_0)) ^ 2.0)))) <= -1e-202)
		tmp = Float64(Float64(J + J) * Float64(Float64(abs(U) / J) * -0.5));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * J) * t_0)) ^ 2.0)))) <= -1e-202)
		tmp = (J + J) * ((abs(U) / J) * -0.5);
	else
		tmp = 2.0 * (0.5 * abs(U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-202], N[(N[(J + J), $MachinePrecision] * N[(N[(N[Abs[U], $MachinePrecision] / J), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(0.5 * N[Abs[U], $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))))) < -1e-202

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 19.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 19.9%

       \[\leadsto 2 \cdot \left(J \cdot \left(-0.5 \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. count-2-rev N/A

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

       5. lift-+.f64 N/A

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

       6. lower-*.f64 19.9%

          \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{U}{J}\right)} \]

   12. Applied rewrites 19.9%

       \[\leadsto \left(J + J\right) \cdot \color{blue}{\left(\frac{U}{J} \cdot -0.5\right)} \]

   if -1e-202 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. 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 12.5%

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

   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 12.6%

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in J around 0

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
   9. 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 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in K around 0

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 27.2%

          \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

   13. Applied rewrites 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 27.2% accurate, 16.6× speedup

Math

\[2 \cdot \left(0.5 \cdot U\right) \]

FPCore

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

C

double code(double J, double K, double U) {
	return 2.0 * (0.5 * U);
}

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 * (0.5d0 * u)
end function

Java

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

Python

def code(J, K, U):
	return 2.0 * (0.5 * U)

Julia

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

MATLAB

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

Wolfram

code[J_, K_, U_] := N[(2.0 * N[(0.5 * U), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

2. 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 12.5%

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

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 12.6%

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

7. Applied rewrites 12.6%

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

8. Taylor expanded in J around 0

   \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
9. 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 27.3%

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

10. Applied rewrites 27.3%

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

11. Taylor expanded in K around 0

    \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
12. Step-by-step derivation

    1. lower-*.f64 27.2%

       \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]

13. Applied rewrites 27.2%

    \[\leadsto 2 \cdot \left(0.5 \cdot U\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025214 
(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)))))