Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 98.8%

Time: 7.1s

Alternatives: 12

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := -1 \cdot \frac{\left|U\right| \cdot t\_0}{\left|t\_0\right|}\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := 2 \cdot \left|J\right|\\ t_5 := \left(t\_1 \cdot t\_3\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{t\_4 \cdot t\_3}\right)}^{2}}\\ t_6 := \cos \left(K \cdot 0.5\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;\left(t\_1 \cdot t\_6\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{t\_4 \cdot t\_6}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (* -2.0 (fabs J)))
        (t_2 (* -1.0 (/ (* (fabs U) t_0) (fabs t_0))))
        (t_3 (cos (/ K 2.0)))
        (t_4 (* 2.0 (fabs J)))
        (t_5 (* (* t_1 t_3) (sqrt (+ 1.0 (pow (/ (fabs U) (* t_4 t_3)) 2.0)))))
        (t_6 (cos (* K 0.5))))
   (*
    (copysign 1.0 J)
    (if (<= t_5 (- INFINITY))
      t_2
      (if (<= t_5 2e+284)
        (* (* t_1 t_6) (sqrt (+ 1.0 (pow (/ (fabs U) (* t_4 t_6)) 2.0))))
        t_2)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = -2.0 * fabs(J);
	double t_2 = -1.0 * ((fabs(U) * t_0) / fabs(t_0));
	double t_3 = cos((K / 2.0));
	double t_4 = 2.0 * fabs(J);
	double t_5 = (t_1 * t_3) * sqrt((1.0 + pow((fabs(U) / (t_4 * t_3)), 2.0)));
	double t_6 = cos((K * 0.5));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_5 <= 2e+284) {
		tmp = (t_1 * t_6) * sqrt((1.0 + pow((fabs(U) / (t_4 * t_6)), 2.0)));
	} else {
		tmp = t_2;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = -2.0 * Math.abs(J);
	double t_2 = -1.0 * ((Math.abs(U) * t_0) / Math.abs(t_0));
	double t_3 = Math.cos((K / 2.0));
	double t_4 = 2.0 * Math.abs(J);
	double t_5 = (t_1 * t_3) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / (t_4 * t_3)), 2.0)));
	double t_6 = Math.cos((K * 0.5));
	double tmp;
	if (t_5 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_5 <= 2e+284) {
		tmp = (t_1 * t_6) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / (t_4 * t_6)), 2.0)));
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	t_1 = -2.0 * math.fabs(J)
	t_2 = -1.0 * ((math.fabs(U) * t_0) / math.fabs(t_0))
	t_3 = math.cos((K / 2.0))
	t_4 = 2.0 * math.fabs(J)
	t_5 = (t_1 * t_3) * math.sqrt((1.0 + math.pow((math.fabs(U) / (t_4 * t_3)), 2.0)))
	t_6 = math.cos((K * 0.5))
	tmp = 0
	if t_5 <= -math.inf:
		tmp = t_2
	elif t_5 <= 2e+284:
		tmp = (t_1 * t_6) * math.sqrt((1.0 + math.pow((math.fabs(U) / (t_4 * t_6)), 2.0)))
	else:
		tmp = t_2
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(-2.0 * abs(J))
	t_2 = Float64(-1.0 * Float64(Float64(abs(U) * t_0) / abs(t_0)))
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(2.0 * abs(J))
	t_5 = Float64(Float64(t_1 * t_3) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(t_4 * t_3)) ^ 2.0))))
	t_6 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_5 <= 2e+284)
		tmp = Float64(Float64(t_1 * t_6) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(t_4 * t_6)) ^ 2.0))));
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K));
	t_1 = -2.0 * abs(J);
	t_2 = -1.0 * ((abs(U) * t_0) / abs(t_0));
	t_3 = cos((K / 2.0));
	t_4 = 2.0 * abs(J);
	t_5 = (t_1 * t_3) * sqrt((1.0 + ((abs(U) / (t_4 * t_3)) ^ 2.0)));
	t_6 = cos((K * 0.5));
	tmp = 0.0;
	if (t_5 <= -Inf)
		tmp = t_2;
	elseif (t_5 <= 2e+284)
		tmp = (t_1 * t_6) * sqrt((1.0 + ((abs(U) / (t_4 * t_6)) ^ 2.0)));
	else
		tmp = t_2;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(N[(N[Abs[U], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$1 * t$95$3), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, (-Infinity)], t$95$2, If[LessEqual[t$95$5, 2e+284], N[(N[(t$95$1 * t$95$6), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(t$95$4 * t$95$6), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000002e284 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 14.9%

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

   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 26.5%

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

   9. Applied rewrites 26.5%

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

   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))))) < 2.0000000000000002e284

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.3%

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

   3. Applied rewrites 73.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.3%

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

   5. Applied rewrites 73.3%

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

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := -1 \cdot \frac{\left|U\right| \cdot t\_0}{\left|t\_0\right|}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(-2 \cdot \left|J\right|\right) \cdot t\_2\\ t_4 := t\_3 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_5 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;t\_3 \cdot \sqrt{\mathsf{fma}\left(t\_5, \frac{t\_5}{0.5 + 0.5 \cdot \cos K}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (* -1.0 (/ (* (fabs U) t_0) (fabs t_0))))
        (t_2 (cos (/ K 2.0)))
        (t_3 (* (* -2.0 (fabs J)) t_2))
        (t_4
         (*
          t_3
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0)))))
        (t_5 (/ (fabs U) (+ (fabs J) (fabs J)))))
   (*
    (copysign 1.0 J)
    (if (<= t_4 (- INFINITY))
      t_1
      (if (<= t_4 2e+284)
        (* t_3 (sqrt (fma t_5 (/ t_5 (+ 0.5 (* 0.5 (cos K)))) 1.0)))
        t_1)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = -1.0 * ((fabs(U) * t_0) / fabs(t_0));
	double t_2 = cos((K / 2.0));
	double t_3 = (-2.0 * fabs(J)) * t_2;
	double t_4 = t_3 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_5 = fabs(U) / (fabs(J) + fabs(J));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_4 <= 2e+284) {
		tmp = t_3 * sqrt(fma(t_5, (t_5 / (0.5 + (0.5 * cos(K)))), 1.0));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(-1.0 * Float64(Float64(abs(U) * t_0) / abs(t_0)))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(-2.0 * abs(J)) * t_2)
	t_4 = Float64(t_3 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_5 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_4 <= 2e+284)
		tmp = Float64(t_3 * sqrt(fma(t_5, Float64(t_5 / Float64(0.5 + Float64(0.5 * cos(K)))), 1.0)));
	else
		tmp = t_1;
	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[(-1.0 * N[(N[(N[Abs[U], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], t$95$1, If[LessEqual[t$95$4, 2e+284], N[(t$95$3 * N[Sqrt[N[(t$95$5 * N[(t$95$5 / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000002e284 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 14.9%

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

   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 26.5%

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

   9. Applied rewrites 26.5%

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

   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))))) < 2.0000000000000002e284

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 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. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 73.2%

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

Alternative 3: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := -1 \cdot \frac{\left|U\right| \cdot t\_0}{\left|t\_0\right|}\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(t\_1 \cdot t\_3\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_3}\right)}^{2}}\\ t_5 := \left|J\right| + \left|J\right|\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;\left(t\_1 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{\left|U\right|}{t\_5}, \frac{\left|U\right|}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot t\_5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (* -2.0 (fabs J)))
        (t_2 (* -1.0 (/ (* (fabs U) t_0) (fabs t_0))))
        (t_3 (cos (/ K 2.0)))
        (t_4
         (*
          (* t_1 t_3)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_3)) 2.0)))))
        (t_5 (+ (fabs J) (fabs J))))
   (*
    (copysign 1.0 J)
    (if (<= t_4 (- INFINITY))
      t_2
      (if (<= t_4 2e+284)
        (*
         (* t_1 (cos (* 0.5 K)))
         (sqrt
          (fma
           (/ (fabs U) t_5)
           (/ (fabs U) (* (fma (cos K) 0.5 0.5) t_5))
           1.0)))
        t_2)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = -2.0 * fabs(J);
	double t_2 = -1.0 * ((fabs(U) * t_0) / fabs(t_0));
	double t_3 = cos((K / 2.0));
	double t_4 = (t_1 * t_3) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_3)), 2.0)));
	double t_5 = fabs(J) + fabs(J);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_4 <= 2e+284) {
		tmp = (t_1 * cos((0.5 * K))) * sqrt(fma((fabs(U) / t_5), (fabs(U) / (fma(cos(K), 0.5, 0.5) * t_5)), 1.0));
	} else {
		tmp = t_2;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(-2.0 * abs(J))
	t_2 = Float64(-1.0 * Float64(Float64(abs(U) * t_0) / abs(t_0)))
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(t_1 * t_3) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_3)) ^ 2.0))))
	t_5 = Float64(abs(J) + abs(J))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_4 <= 2e+284)
		tmp = Float64(Float64(t_1 * cos(Float64(0.5 * K))) * sqrt(fma(Float64(abs(U) / t_5), Float64(abs(U) / Float64(fma(cos(K), 0.5, 0.5) * t_5)), 1.0)));
	else
		tmp = t_2;
	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[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(N[(N[Abs[U], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 * t$95$3), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[J], $MachinePrecision] + 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$4, (-Infinity)], t$95$2, If[LessEqual[t$95$4, 2e+284], N[(N[(t$95$1 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / t$95$5), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000002e284 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 14.9%

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

   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 26.5%

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

   9. Applied rewrites 26.5%

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

   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))))) < 2.0000000000000002e284

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 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. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 73.2%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 73.2%

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lift-+.f64 73.2%

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in K around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 73.2%

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

   8. Applied rewrites 73.2%

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

Alternative 4: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := -1 \cdot \frac{\left|U\right| \cdot t\_0}{\left|t\_0\right|}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left|J\right| + \left|J\right|\\ t_4 := \left(-2 \cdot \left|J\right|\right) \cdot t\_2\\ t_5 := t\_4 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_6 := \frac{\left|U\right|}{t\_3}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq -5 \cdot 10^{-54}:\\ \;\;\;\;\left(\left(t\_0 \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{\left|U\right|}{\left(t\_3 \cdot t\_3\right) \cdot \left(0.5 + 0.5 \cdot \cos K\right)}, \left|U\right|, 1\right)}\right) \cdot \left|J\right|\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;t\_4 \cdot \sqrt{\mathsf{fma}\left(t\_6, \frac{t\_6}{0.5 + 0.5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (* -1.0 (/ (* (fabs U) t_0) (fabs t_0))))
        (t_2 (cos (/ K 2.0)))
        (t_3 (+ (fabs J) (fabs J)))
        (t_4 (* (* -2.0 (fabs J)) t_2))
        (t_5
         (*
          t_4
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0)))))
        (t_6 (/ (fabs U) t_3)))
   (*
    (copysign 1.0 J)
    (if (<= t_5 (- INFINITY))
      t_1
      (if (<= t_5 -5e-54)
        (*
         (*
          (* t_0 -2.0)
          (sqrt
           (fma
            (/ (fabs U) (* (* t_3 t_3) (+ 0.5 (* 0.5 (cos K)))))
            (fabs U)
            1.0)))
         (fabs J))
        (if (<= t_5 2e+284)
          (* t_4 (sqrt (fma t_6 (/ t_6 (+ 0.5 0.5)) 1.0)))
          t_1))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = -1.0 * ((fabs(U) * t_0) / fabs(t_0));
	double t_2 = cos((K / 2.0));
	double t_3 = fabs(J) + fabs(J);
	double t_4 = (-2.0 * fabs(J)) * t_2;
	double t_5 = t_4 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_6 = fabs(U) / t_3;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_5 <= -5e-54) {
		tmp = ((t_0 * -2.0) * sqrt(fma((fabs(U) / ((t_3 * t_3) * (0.5 + (0.5 * cos(K))))), fabs(U), 1.0))) * fabs(J);
	} else if (t_5 <= 2e+284) {
		tmp = t_4 * sqrt(fma(t_6, (t_6 / (0.5 + 0.5)), 1.0));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(-1.0 * Float64(Float64(abs(U) * t_0) / abs(t_0)))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(abs(J) + abs(J))
	t_4 = Float64(Float64(-2.0 * abs(J)) * t_2)
	t_5 = Float64(t_4 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_6 = Float64(abs(U) / t_3)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_5 <= -5e-54)
		tmp = Float64(Float64(Float64(t_0 * -2.0) * sqrt(fma(Float64(abs(U) / Float64(Float64(t_3 * t_3) * Float64(0.5 + Float64(0.5 * cos(K))))), abs(U), 1.0))) * abs(J));
	elseif (t_5 <= 2e+284)
		tmp = Float64(t_4 * sqrt(fma(t_6, Float64(t_6 / Float64(0.5 + 0.5)), 1.0)));
	else
		tmp = t_1;
	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[(-1.0 * N[(N[(N[Abs[U], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[U], $MachinePrecision] / t$95$3), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, (-Infinity)], t$95$1, If[LessEqual[t$95$5, -5e-54], N[(N[(N[(t$95$0 * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(t$95$3 * t$95$3), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2e+284], N[(t$95$4 * N[Sqrt[N[(t$95$6 * N[(t$95$6 / N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000002e284 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 14.9%

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

   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 26.5%

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

   9. Applied rewrites 26.5%

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

   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))))) < -5.0000000000000002e-54

   1. Initial program 73.3%

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

   3. Applied rewrites 61.6%

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

   if -5.0000000000000002e-54 < (*.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))))) < 2.0000000000000002e284

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 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. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 63.8%

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

Alternative 5: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := -1 \cdot \frac{\left|U\right| \cdot t\_0}{\left|t\_0\right|}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(-2 \cdot \left|J\right|\right) \cdot t\_2\\ t_4 := t\_3 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_5 := \left|J\right| + \left|J\right|\\ t_6 := \frac{\left|U\right|}{t\_5}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-54}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{\left|U\right|}{\left(t\_5 \cdot t\_5\right) \cdot \left(0.5 + 0.5 \cdot \cos K\right)}, \left|U\right|, 1\right)} \cdot \left(\left|J\right| \cdot -2\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;t\_3 \cdot \sqrt{\mathsf{fma}\left(t\_6, \frac{t\_6}{0.5 + 0.5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (* -1.0 (/ (* (fabs U) t_0) (fabs t_0))))
        (t_2 (cos (/ K 2.0)))
        (t_3 (* (* -2.0 (fabs J)) t_2))
        (t_4
         (*
          t_3
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0)))))
        (t_5 (+ (fabs J) (fabs J)))
        (t_6 (/ (fabs U) t_5)))
   (*
    (copysign 1.0 J)
    (if (<= t_4 (- INFINITY))
      t_1
      (if (<= t_4 -5e-54)
        (*
         (*
          (sqrt
           (fma
            (/ (fabs U) (* (* t_5 t_5) (+ 0.5 (* 0.5 (cos K)))))
            (fabs U)
            1.0))
          (* (fabs J) -2.0))
         t_0)
        (if (<= t_4 2e+284)
          (* t_3 (sqrt (fma t_6 (/ t_6 (+ 0.5 0.5)) 1.0)))
          t_1))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = -1.0 * ((fabs(U) * t_0) / fabs(t_0));
	double t_2 = cos((K / 2.0));
	double t_3 = (-2.0 * fabs(J)) * t_2;
	double t_4 = t_3 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_5 = fabs(J) + fabs(J);
	double t_6 = fabs(U) / t_5;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_4 <= -5e-54) {
		tmp = (sqrt(fma((fabs(U) / ((t_5 * t_5) * (0.5 + (0.5 * cos(K))))), fabs(U), 1.0)) * (fabs(J) * -2.0)) * t_0;
	} else if (t_4 <= 2e+284) {
		tmp = t_3 * sqrt(fma(t_6, (t_6 / (0.5 + 0.5)), 1.0));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(-1.0 * Float64(Float64(abs(U) * t_0) / abs(t_0)))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(-2.0 * abs(J)) * t_2)
	t_4 = Float64(t_3 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_5 = Float64(abs(J) + abs(J))
	t_6 = Float64(abs(U) / t_5)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_4 <= -5e-54)
		tmp = Float64(Float64(sqrt(fma(Float64(abs(U) / Float64(Float64(t_5 * t_5) * Float64(0.5 + Float64(0.5 * cos(K))))), abs(U), 1.0)) * Float64(abs(J) * -2.0)) * t_0);
	elseif (t_4 <= 2e+284)
		tmp = Float64(t_3 * sqrt(fma(t_6, Float64(t_6 / Float64(0.5 + 0.5)), 1.0)));
	else
		tmp = t_1;
	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[(-1.0 * N[(N[(N[Abs[U], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[U], $MachinePrecision] / t$95$5), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], t$95$1, If[LessEqual[t$95$4, -5e-54], N[(N[(N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(t$95$5 * t$95$5), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$4, 2e+284], N[(t$95$3 * N[Sqrt[N[(t$95$6 * N[(t$95$6 / N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000002e284 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 14.9%

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

   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 26.5%

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

   9. Applied rewrites 26.5%

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

   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))))) < -5.0000000000000002e-54

   1. Initial program 73.3%

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

   3. Applied rewrites 61.6%

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

   if -5.0000000000000002e-54 < (*.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))))) < 2.0000000000000002e284

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 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. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 63.8%

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

Alternative 6: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := -1 \cdot \frac{\left|U\right| \cdot t\_0}{\left|t\_0\right|}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(-2 \cdot \left|J\right|\right) \cdot t\_2\\ t_4 := t\_3 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_5 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;t\_3 \cdot \sqrt{\mathsf{fma}\left(t\_5, \frac{t\_5}{0.5 + 0.5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (* -1.0 (/ (* (fabs U) t_0) (fabs t_0))))
        (t_2 (cos (/ K 2.0)))
        (t_3 (* (* -2.0 (fabs J)) t_2))
        (t_4
         (*
          t_3
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0)))))
        (t_5 (/ (fabs U) (+ (fabs J) (fabs J)))))
   (*
    (copysign 1.0 J)
    (if (<= t_4 (- INFINITY))
      t_1
      (if (<= t_4 2e+284)
        (* t_3 (sqrt (fma t_5 (/ t_5 (+ 0.5 0.5)) 1.0)))
        t_1)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = -1.0 * ((fabs(U) * t_0) / fabs(t_0));
	double t_2 = cos((K / 2.0));
	double t_3 = (-2.0 * fabs(J)) * t_2;
	double t_4 = t_3 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_5 = fabs(U) / (fabs(J) + fabs(J));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_4 <= 2e+284) {
		tmp = t_3 * sqrt(fma(t_5, (t_5 / (0.5 + 0.5)), 1.0));
	} else {
		tmp = t_1;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(-1.0 * Float64(Float64(abs(U) * t_0) / abs(t_0)))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(-2.0 * abs(J)) * t_2)
	t_4 = Float64(t_3 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_5 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_4 <= 2e+284)
		tmp = Float64(t_3 * sqrt(fma(t_5, Float64(t_5 / Float64(0.5 + 0.5)), 1.0)));
	else
		tmp = t_1;
	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[(-1.0 * N[(N[(N[Abs[U], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], t$95$1, If[LessEqual[t$95$4, 2e+284], N[(t$95$3 * N[Sqrt[N[(t$95$5 * N[(t$95$5 / N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000002e284 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 14.9%

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

   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 26.5%

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

   9. Applied rewrites 26.5%

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

   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))))) < 2.0000000000000002e284

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 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. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 63.8%

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

Alternative 7: 63.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{U}{J + J}\\ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, \frac{t\_0}{0.5 + 0.5}, 1\right)} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ U (+ J J))))
   (*
    (* (* -2.0 J) (cos (/ K 2.0)))
    (sqrt (fma t_0 (/ t_0 (+ 0.5 0.5)) 1.0)))))

C

double code(double J, double K, double U) {
	double t_0 = U / (J + J);
	return ((-2.0 * J) * cos((K / 2.0))) * sqrt(fma(t_0, (t_0 / (0.5 + 0.5)), 1.0));
}

Julia

function code(J, K, U)
	t_0 = Float64(U / Float64(J + J))
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * sqrt(fma(t_0, Float64(t_0 / Float64(0.5 + 0.5)), 1.0)))
end

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-/.f64 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. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. frac-times N/A

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

   12. associate-/l* N/A

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

   13. lower-fma.f64 N/A

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

3. Applied rewrites 73.2%

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

4. Taylor expanded in K around 0

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

6. Applied rewrites 63.8%

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

Alternative 8: 54.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ t_2 := -2 \cdot \left(\left|J\right| \cdot \cos \left(-0.5 \cdot K\right)\right)\\ t_3 := {\left(\left|J\right|\right)}^{2}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-56}:\\ \;\;\;\;-2 \cdot \left(\left|J\right| \cdot \sqrt{1 + 0.25 \cdot \frac{{\left(\left|U\right|\right)}^{2}}{t\_3}}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+303}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \sqrt{\frac{0.25}{t\_3}}\right)\right)\\ \end{array} \end{array} \]

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 (* -2.0 (* (fabs J) (cos (* -0.5 K)))))
        (t_3 (pow (fabs J) 2.0)))
   (*
    (copysign 1.0 J)
    (if (<= t_1 -1e+144)
      t_2
      (if (<= t_1 -2e-56)
        (*
         -2.0
         (* (fabs J) (sqrt (+ 1.0 (* 0.25 (/ (pow (fabs U) 2.0) t_3))))))
        (if (<= t_1 1e+303)
          t_2
          (* 2.0 (* (fabs J) (* (fabs U) (sqrt (/ 0.25 t_3)))))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double t_2 = -2.0 * (fabs(J) * cos((-0.5 * K)));
	double t_3 = pow(fabs(J), 2.0);
	double tmp;
	if (t_1 <= -1e+144) {
		tmp = t_2;
	} else if (t_1 <= -2e-56) {
		tmp = -2.0 * (fabs(J) * sqrt((1.0 + (0.25 * (pow(fabs(U), 2.0) / t_3)))));
	} else if (t_1 <= 1e+303) {
		tmp = t_2;
	} else {
		tmp = 2.0 * (fabs(J) * (fabs(U) * sqrt((0.25 / t_3))));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * Math.abs(J)) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_0)), 2.0)));
	double t_2 = -2.0 * (Math.abs(J) * Math.cos((-0.5 * K)));
	double t_3 = Math.pow(Math.abs(J), 2.0);
	double tmp;
	if (t_1 <= -1e+144) {
		tmp = t_2;
	} else if (t_1 <= -2e-56) {
		tmp = -2.0 * (Math.abs(J) * Math.sqrt((1.0 + (0.25 * (Math.pow(Math.abs(U), 2.0) / t_3)))));
	} else if (t_1 <= 1e+303) {
		tmp = t_2;
	} else {
		tmp = 2.0 * (Math.abs(J) * (Math.abs(U) * Math.sqrt((0.25 / t_3))));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * math.fabs(J)) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_0)), 2.0)))
	t_2 = -2.0 * (math.fabs(J) * math.cos((-0.5 * K)))
	t_3 = math.pow(math.fabs(J), 2.0)
	tmp = 0
	if t_1 <= -1e+144:
		tmp = t_2
	elif t_1 <= -2e-56:
		tmp = -2.0 * (math.fabs(J) * math.sqrt((1.0 + (0.25 * (math.pow(math.fabs(U), 2.0) / t_3)))))
	elif t_1 <= 1e+303:
		tmp = t_2
	else:
		tmp = 2.0 * (math.fabs(J) * (math.fabs(U) * math.sqrt((0.25 / t_3))))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * abs(J)) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0))))
	t_2 = Float64(-2.0 * Float64(abs(J) * cos(Float64(-0.5 * K))))
	t_3 = abs(J) ^ 2.0
	tmp = 0.0
	if (t_1 <= -1e+144)
		tmp = t_2;
	elseif (t_1 <= -2e-56)
		tmp = Float64(-2.0 * Float64(abs(J) * sqrt(Float64(1.0 + Float64(0.25 * Float64((abs(U) ^ 2.0) / t_3))))));
	elseif (t_1 <= 1e+303)
		tmp = t_2;
	else
		tmp = Float64(2.0 * Float64(abs(J) * Float64(abs(U) * sqrt(Float64(0.25 / t_3)))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * abs(J)) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_0)) ^ 2.0)));
	t_2 = -2.0 * (abs(J) * cos((-0.5 * K)));
	t_3 = abs(J) ^ 2.0;
	tmp = 0.0;
	if (t_1 <= -1e+144)
		tmp = t_2;
	elseif (t_1 <= -2e-56)
		tmp = -2.0 * (abs(J) * sqrt((1.0 + (0.25 * ((abs(U) ^ 2.0) / t_3)))));
	elseif (t_1 <= 1e+303)
		tmp = t_2;
	else
		tmp = 2.0 * (abs(J) * (abs(U) * sqrt((0.25 / t_3))));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Abs[J], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -1e+144], t$95$2, If[LessEqual[t$95$1, -2e-56], N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[N[Abs[U], $MachinePrecision], 2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+303], t$95$2, N[(2.0 * N[(N[Abs[J], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] * N[Sqrt[N[(0.25 / t$95$3), $MachinePrecision]], $MachinePrecision]), $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))))) < -1e144 or -2.0000000000000001e-56 < (*.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))))) < 1e303

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 62.0%

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

   4. Applied rewrites 62.0%

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 50.9%

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

   9. Applied rewrites 50.9%

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

   if -1e144 < (*.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-56

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 32.6%

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

   4. Applied rewrites 32.6%

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

   if 1e303 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 13.4%

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

   7. Applied rewrites 13.4%

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2}}}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;-2 \cdot \left(\left|J\right| \cdot \cos \left(-0.5 \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    (copysign 1.0 J)
    (if (<=
         (*
          (* (* -2.0 (fabs J)) t_0)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0))))
         1e+303)
      (* -2.0 (* (fabs J) (cos (* -0.5 K))))
      (* 2.0 (* (fabs J) (* (fabs U) (sqrt (/ 0.25 (pow (fabs J) 2.0))))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)))) <= 1e+303) {
		tmp = -2.0 * (fabs(J) * cos((-0.5 * K)));
	} else {
		tmp = 2.0 * (fabs(J) * (fabs(U) * sqrt((0.25 / pow(fabs(J), 2.0)))));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * Math.abs(J)) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_0)), 2.0)))) <= 1e+303) {
		tmp = -2.0 * (Math.abs(J) * Math.cos((-0.5 * K)));
	} else {
		tmp = 2.0 * (Math.abs(J) * (Math.abs(U) * Math.sqrt((0.25 / Math.pow(Math.abs(J), 2.0)))));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * math.fabs(J)) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_0)), 2.0)))) <= 1e+303:
		tmp = -2.0 * (math.fabs(J) * math.cos((-0.5 * K)))
	else:
		tmp = 2.0 * (math.fabs(J) * (math.fabs(U) * math.sqrt((0.25 / math.pow(math.fabs(J), 2.0)))))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * abs(J)) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0)))) <= 1e+303)
		tmp = Float64(-2.0 * Float64(abs(J) * cos(Float64(-0.5 * K))));
	else
		tmp = Float64(2.0 * Float64(abs(J) * Float64(abs(U) * sqrt(Float64(0.25 / (abs(J) ^ 2.0))))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * abs(J)) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_0)) ^ 2.0)))) <= 1e+303)
		tmp = -2.0 * (abs(J) * cos((-0.5 * K)));
	else
		tmp = 2.0 * (abs(J) * (abs(U) * sqrt((0.25 / (abs(J) ^ 2.0)))));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+303], N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Abs[J], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] * N[Sqrt[N[(0.25 / N[Power[N[Abs[J], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $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))))) < 1e303

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 62.0%

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

   4. Applied rewrites 62.0%

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 50.9%

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

   9. Applied rewrites 50.9%

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

   if 1e303 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 13.4%

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

   7. Applied rewrites 13.4%

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2}}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 32.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}} \leq 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(\left|J\right|, -2, \left(K \cdot K\right) \cdot \mathsf{fma}\left(\left(K \cdot K\right) \cdot -0.005208333333333333, \left|J\right|, 0.25 \cdot \left|J\right|\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    (copysign 1.0 J)
    (if (<=
         (*
          (* (* -2.0 (fabs J)) t_0)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0))))
         1e-153)
      (*
       (fma
        (fabs J)
        -2.0
        (*
         (* K K)
         (fma (* (* K K) -0.005208333333333333) (fabs J) (* 0.25 (fabs J)))))
       1.0)
      (* 2.0 (* (fabs J) (* (fabs U) (sqrt (/ 0.25 (pow (fabs J) 2.0))))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)))) <= 1e-153) {
		tmp = fma(fabs(J), -2.0, ((K * K) * fma(((K * K) * -0.005208333333333333), fabs(J), (0.25 * fabs(J))))) * 1.0;
	} else {
		tmp = 2.0 * (fabs(J) * (fabs(U) * sqrt((0.25 / pow(fabs(J), 2.0)))));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * abs(J)) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0)))) <= 1e-153)
		tmp = Float64(fma(abs(J), -2.0, Float64(Float64(K * K) * fma(Float64(Float64(K * K) * -0.005208333333333333), abs(J), Float64(0.25 * abs(J))))) * 1.0);
	else
		tmp = Float64(2.0 * Float64(abs(J) * Float64(abs(U) * sqrt(Float64(0.25 / (abs(J) ^ 2.0))))));
	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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-153], N[(N[(N[Abs[J], $MachinePrecision] * -2.0 + N[(N[(K * K), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.005208333333333333), $MachinePrecision] * N[Abs[J], $MachinePrecision] + N[(0.25 * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(2.0 * N[(N[Abs[J], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] * N[Sqrt[N[(0.25 / N[Power[N[Abs[J], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $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-153

   1. Initial program 73.3%

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

   2. Taylor expanded in J around inf

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

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 26.9%

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

   7. Applied rewrites 26.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-+l+ N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-fma.f64 N/A

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

   9. Applied rewrites 26.0%

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

   11. Applied rewrites 26.9%

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

   if 1e-153 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 13.4%

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

   7. Applied rewrites 13.4%

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2}}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 27.0% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (fma J -2.0 (* (* K K) (fma (* (* K K) -0.005208333333333333) J (* 0.25 J))))
  1.0))

C

double code(double J, double K, double U) {
	return fma(J, -2.0, ((K * K) * fma(((K * K) * -0.005208333333333333), J, (0.25 * J)))) * 1.0;
}

Julia

function code(J, K, U)
	return Float64(fma(J, -2.0, Float64(Float64(K * K) * fma(Float64(Float64(K * K) * -0.005208333333333333), J, Float64(0.25 * J)))) * 1.0)
end

Wolfram

code[J_, K_, U_] := N[(N[(J * -2.0 + N[(N[(K * K), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.005208333333333333), $MachinePrecision] * J + N[(0.25 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in J around inf

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in K around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 26.9%

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

7. Applied rewrites 26.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-+l+ N/A

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lift-fma.f64 N/A

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

9. Applied rewrites 26.0%

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

11. Applied rewrites 26.9%

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

Alternative 12: 26.9% accurate, 6.5× speedup

Math

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

FPCore

(FPCore (J K U)
 :precision binary64
 (* (fma (* (* 0.25 J) K) K (* J -2.0)) 1.0))

C

double code(double J, double K, double U) {
	return fma(((0.25 * J) * K), K, (J * -2.0)) * 1.0;
}

Julia

function code(J, K, U)
	return Float64(fma(Float64(Float64(0.25 * J) * K), K, Float64(J * -2.0)) * 1.0)
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(N[(0.25 * J), $MachinePrecision] * K), $MachinePrecision] * K + N[(J * -2.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in J around inf

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in K around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 27.0%

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

7. Applied rewrites 27.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 27.0%

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 27.0%

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

9. Applied rewrites 27.0%

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

Reproduce

herbie shell --seed 2025205 
(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)))))