Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.6% → 99.5%

Time: 6.7s

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: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := -2 \cdot \left(\left|U\right| \cdot \left(t\_1 \cdot \sqrt{\frac{0.25}{{t\_1}^{2}}}\right)\right)\\ t_3 := 2 \cdot \left|J\right|\\ t_4 := -2 \cdot \left|J\right|\\ t_5 := \left(t\_4 \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{t\_3 \cdot t\_0}\right)}^{2}}\\ t_6 := \cos \left(K \cdot 0.5\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left(t\_4 \cdot t\_6\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{t\_3 \cdot t\_6}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (cos (* 0.5 K)))
        (t_2 (* -2.0 (* (fabs U) (* t_1 (sqrt (/ 0.25 (pow t_1 2.0)))))))
        (t_3 (* 2.0 (fabs J)))
        (t_4 (* -2.0 (fabs J)))
        (t_5 (* (* t_4 t_0) (sqrt (+ 1.0 (pow (/ (fabs U) (* t_3 t_0)) 2.0)))))
        (t_6 (cos (* K 0.5))))
   (*
    (copysign 1.0 J)
    (if (<= t_5 (- INFINITY))
      t_2
      (if (<= t_5 2e+301)
        (* (* t_4 t_6) (sqrt (+ 1.0 (pow (/ (fabs U) (* t_3 t_6)) 2.0))))
        t_2)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = cos((0.5 * K));
	double t_2 = -2.0 * (fabs(U) * (t_1 * sqrt((0.25 / pow(t_1, 2.0)))));
	double t_3 = 2.0 * fabs(J);
	double t_4 = -2.0 * fabs(J);
	double t_5 = (t_4 * t_0) * sqrt((1.0 + pow((fabs(U) / (t_3 * t_0)), 2.0)));
	double t_6 = cos((K * 0.5));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_5 <= 2e+301) {
		tmp = (t_4 * t_6) * sqrt((1.0 + pow((fabs(U) / (t_3 * t_6)), 2.0)));
	} else {
		tmp = t_2;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.cos((0.5 * K));
	double t_2 = -2.0 * (Math.abs(U) * (t_1 * Math.sqrt((0.25 / Math.pow(t_1, 2.0)))));
	double t_3 = 2.0 * Math.abs(J);
	double t_4 = -2.0 * Math.abs(J);
	double t_5 = (t_4 * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / (t_3 * t_0)), 2.0)));
	double t_6 = Math.cos((K * 0.5));
	double tmp;
	if (t_5 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_5 <= 2e+301) {
		tmp = (t_4 * t_6) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / (t_3 * t_6)), 2.0)));
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.cos((0.5 * K))
	t_2 = -2.0 * (math.fabs(U) * (t_1 * math.sqrt((0.25 / math.pow(t_1, 2.0)))))
	t_3 = 2.0 * math.fabs(J)
	t_4 = -2.0 * math.fabs(J)
	t_5 = (t_4 * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / (t_3 * t_0)), 2.0)))
	t_6 = math.cos((K * 0.5))
	tmp = 0
	if t_5 <= -math.inf:
		tmp = t_2
	elif t_5 <= 2e+301:
		tmp = (t_4 * t_6) * math.sqrt((1.0 + math.pow((math.fabs(U) / (t_3 * t_6)), 2.0)))
	else:
		tmp = t_2
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = cos(Float64(0.5 * K))
	t_2 = Float64(-2.0 * Float64(abs(U) * Float64(t_1 * sqrt(Float64(0.25 / (t_1 ^ 2.0))))))
	t_3 = Float64(2.0 * abs(J))
	t_4 = Float64(-2.0 * abs(J))
	t_5 = Float64(Float64(t_4 * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(t_3 * t_0)) ^ 2.0))))
	t_6 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_5 <= 2e+301)
		tmp = Float64(Float64(t_4 * t_6) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(t_3 * t_6)) ^ 2.0))));
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = cos((0.5 * K));
	t_2 = -2.0 * (abs(U) * (t_1 * sqrt((0.25 / (t_1 ^ 2.0)))));
	t_3 = 2.0 * abs(J);
	t_4 = -2.0 * abs(J);
	t_5 = (t_4 * t_0) * sqrt((1.0 + ((abs(U) / (t_3 * t_0)) ^ 2.0)));
	t_6 = cos((K * 0.5));
	tmp = 0.0;
	if (t_5 <= -Inf)
		tmp = t_2;
	elseif (t_5 <= 2e+301)
		tmp = (t_4 * t_6) * sqrt((1.0 + ((abs(U) / (t_3 * t_6)) ^ 2.0)));
	else
		tmp = t_2;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(N[Abs[U], $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(0.25 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, (-Infinity)], t$95$2, If[LessEqual[t$95$5, 2e+301], N[(N[(t$95$4 * t$95$6), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(t$95$3 * t$95$6), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000001e301 < (*.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 73.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 12.9%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 27.1%

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

   7. Applied rewrites 27.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e301

   1. Initial program 73.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 73.6%

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

   3. Applied rewrites 73.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 73.6%

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

   5. Applied rewrites 73.6%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(0.5 \cdot K\right)\\ t_3 := -2 \cdot \left(\left|U\right| \cdot \left(t\_2 \cdot \sqrt{\frac{0.25}{{t\_2}^{2}}}\right)\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{\left|U\right|}{\left|J\right|}, \frac{\left|U\right|}{\left(\left(\cos K - -1\right) \cdot 2\right) \cdot \left|J\right|}, 1\right)} \cdot \left|J\right|\right) \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 (fabs J)) t_0)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0)))))
        (t_2 (cos (* 0.5 K)))
        (t_3 (* -2.0 (* (fabs U) (* t_2 (sqrt (/ 0.25 (pow t_2 2.0))))))))
   (*
    (copysign 1.0 J)
    (if (<= t_1 (- INFINITY))
      t_3
      (if (<= t_1 2e+301)
        (*
         (*
          (sqrt
           (fma
            (/ (fabs U) (fabs J))
            (/ (fabs U) (* (* (- (cos K) -1.0) 2.0) (fabs J)))
            1.0))
          (fabs J))
         (* (cos (* -0.5 K)) -2.0))
        t_3)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double t_2 = cos((0.5 * K));
	double t_3 = -2.0 * (fabs(U) * (t_2 * sqrt((0.25 / pow(t_2, 2.0)))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 2e+301) {
		tmp = (sqrt(fma((fabs(U) / fabs(J)), (fabs(U) / (((cos(K) - -1.0) * 2.0) * fabs(J))), 1.0)) * fabs(J)) * (cos((-0.5 * K)) * -2.0);
	} else {
		tmp = t_3;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * abs(J)) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	t_3 = Float64(-2.0 * Float64(abs(U) * Float64(t_2 * sqrt(Float64(0.25 / (t_2 ^ 2.0))))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+301)
		tmp = Float64(Float64(sqrt(fma(Float64(abs(U) / abs(J)), Float64(abs(U) / Float64(Float64(Float64(cos(K) - -1.0) * 2.0) * abs(J))), 1.0)) * abs(J)) * Float64(cos(Float64(-0.5 * K)) * -2.0));
	else
		tmp = t_3;
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[(N[Abs[U], $MachinePrecision] * N[(t$95$2 * N[Sqrt[N[(0.25 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+301], N[(N[(N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[(N[(N[Cos[K], $MachinePrecision] - -1.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], t$95$3]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000001e301 < (*.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 73.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 12.9%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 27.1%

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

   7. Applied rewrites 27.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e301

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.4%

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. distribute-lft1-in N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 73.4%

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

   7. Applied rewrites 73.4%

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

Alternative 3: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right) \cdot -2\\ t_1 := \left(\left|U\right| \cdot \sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos K}}\right) \cdot t\_0\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{\left|U\right|}{\left|J\right|}, \frac{\left|U\right|}{\left(\left(\cos K - -1\right) \cdot 2\right) \cdot \left|J\right|}, 1\right)} \cdot \left|J\right|\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (cos (* -0.5 K)) -2.0))
        (t_1 (* (* (fabs U) (sqrt (/ 0.25 (+ 0.5 (* 0.5 (cos K)))))) t_0))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 (fabs J)) t_2)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_3 (- INFINITY))
      t_1
      (if (<= t_3 2e+301)
        (*
         (*
          (sqrt
           (fma
            (/ (fabs U) (fabs J))
            (/ (fabs U) (* (* (- (cos K) -1.0) 2.0) (fabs J)))
            1.0))
          (fabs J))
         t_0)
        t_1)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K)) * -2.0;
	double t_1 = (fabs(U) * sqrt((0.25 / (0.5 + (0.5 * cos(K)))))) * t_0;
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * fabs(J)) * t_2) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= 2e+301) {
		tmp = (sqrt(fma((fabs(U) / fabs(J)), (fabs(U) / (((cos(K) - -1.0) * 2.0) * fabs(J))), 1.0)) * fabs(J)) * t_0;
	} else {
		tmp = t_1;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(cos(Float64(-0.5 * K)) * -2.0)
	t_1 = Float64(Float64(abs(U) * sqrt(Float64(0.25 / Float64(0.5 + Float64(0.5 * cos(K)))))) * t_0)
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * abs(J)) * t_2) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= 2e+301)
		tmp = Float64(Float64(sqrt(fma(Float64(abs(U) / abs(J)), Float64(abs(U) / Float64(Float64(Float64(cos(K) - -1.0) * 2.0) * abs(J))), 1.0)) * abs(J)) * t_0);
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[U], $MachinePrecision] * N[Sqrt[N[(0.25 / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, 2e+301], N[(N[(N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[(N[(N[Cos[K], $MachinePrecision] - -1.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $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 2.0000000000000001e301 < (*.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 73.6%

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 14.7%

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

   6. Applied rewrites 14.7%

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

   7. Taylor expanded in U around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 27.0%

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

   9. Applied rewrites 27.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e301

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 73.4%

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. distribute-lft1-in N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 73.4%

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

   7. Applied rewrites 73.4%

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

Alternative 4: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|U\right| \cdot \sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos K}}\right) \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot -2\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot \left|J\right|\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + {\left(0.5 \cdot \frac{\left|U\right|}{\left|J\right|}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0
         (*
          (* (fabs U) (sqrt (/ 0.25 (+ 0.5 (* 0.5 (cos K))))))
          (* (cos (* -0.5 K)) -2.0)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 (fabs J)) t_1))
        (t_3
         (*
          t_2
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_3 (- INFINITY))
      t_0
      (if (<= t_3 2e+301)
        (* t_2 (sqrt (+ 1.0 (pow (* 0.5 (/ (fabs U) (fabs J))) 2.0))))
        t_0)))))

C

double code(double J, double K, double U) {
	double t_0 = (fabs(U) * sqrt((0.25 / (0.5 + (0.5 * cos(K)))))) * (cos((-0.5 * K)) * -2.0);
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * fabs(J)) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_3 <= 2e+301) {
		tmp = t_2 * sqrt((1.0 + pow((0.5 * (fabs(U) / fabs(J))), 2.0)));
	} else {
		tmp = t_0;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = (Math.abs(U) * Math.sqrt((0.25 / (0.5 + (0.5 * Math.cos(K)))))) * (Math.cos((-0.5 * K)) * -2.0);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = (-2.0 * Math.abs(J)) * t_1;
	double t_3 = t_2 * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_3 <= 2e+301) {
		tmp = t_2 * Math.sqrt((1.0 + Math.pow((0.5 * (Math.abs(U) / Math.abs(J))), 2.0)));
	} else {
		tmp = t_0;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = (math.fabs(U) * math.sqrt((0.25 / (0.5 + (0.5 * math.cos(K)))))) * (math.cos((-0.5 * K)) * -2.0)
	t_1 = math.cos((K / 2.0))
	t_2 = (-2.0 * math.fabs(J)) * t_1
	t_3 = t_2 * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_0
	elif t_3 <= 2e+301:
		tmp = t_2 * math.sqrt((1.0 + math.pow((0.5 * (math.fabs(U) / math.fabs(J))), 2.0)))
	else:
		tmp = t_0
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(Float64(abs(U) * sqrt(Float64(0.25 / Float64(0.5 + Float64(0.5 * cos(K)))))) * Float64(cos(Float64(-0.5 * K)) * -2.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * abs(J)) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_3 <= 2e+301)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(abs(U) / abs(J))) ^ 2.0))));
	else
		tmp = t_0;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = (abs(U) * sqrt((0.25 / (0.5 + (0.5 * cos(K)))))) * (cos((-0.5 * K)) * -2.0);
	t_1 = cos((K / 2.0));
	t_2 = (-2.0 * abs(J)) * t_1;
	t_3 = t_2 * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_0;
	elseif (t_3 <= 2e+301)
		tmp = t_2 * sqrt((1.0 + ((0.5 * (abs(U) / abs(J))) ^ 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(N[Abs[U], $MachinePrecision] * N[Sqrt[N[(0.25 / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], t$95$0, If[LessEqual[t$95$3, 2e+301], N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]), $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 2.0000000000000001e301 < (*.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 73.6%

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 14.7%

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

   6. Applied rewrites 14.7%

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

   7. Taylor expanded in U around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 27.0%

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

   9. Applied rewrites 27.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e301

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.4%

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

   4. Applied rewrites 64.4%

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

Alternative 5: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|U\right|}{\left|J\right|}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\sqrt{0.25 \cdot \frac{{\left(\left|U\right|\right)}^{2}}{0.5 + 0.5 \cdot \cos K}} \cdot -2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(t\_0, 0.25 \cdot t\_0, 1\right)} \cdot \left|J\right|\right) \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ (fabs U) (fabs J)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 (fabs J)) t_1)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_2 (- INFINITY))
      (* (sqrt (* 0.25 (/ (pow (fabs U) 2.0) (+ 0.5 (* 0.5 (cos K)))))) -2.0)
      (if (<= t_2 2e+301)
        (*
         (* (sqrt (fma t_0 (* 0.25 t_0) 1.0)) (fabs J))
         (* (cos (* -0.5 K)) -2.0))
        (*
         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 = fabs(U) / fabs(J);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * fabs(J)) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sqrt((0.25 * (pow(fabs(U), 2.0) / (0.5 + (0.5 * cos(K)))))) * -2.0;
	} else if (t_2 <= 2e+301) {
		tmp = (sqrt(fma(t_0, (0.25 * t_0), 1.0)) * fabs(J)) * (cos((-0.5 * K)) * -2.0);
	} 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 = Float64(abs(U) / abs(J))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * abs(J)) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(0.25 * Float64((abs(U) ^ 2.0) / Float64(0.5 + Float64(0.5 * cos(K)))))) * -2.0);
	elseif (t_2 <= 2e+301)
		tmp = Float64(Float64(sqrt(fma(t_0, Float64(0.25 * t_0), 1.0)) * abs(J)) * Float64(cos(Float64(-0.5 * K)) * -2.0));
	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[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(N[Sqrt[N[(0.25 * N[(N[Power[N[Abs[U], $MachinePrecision], 2.0], $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+301], N[(N[(N[Sqrt[N[(t$95$0 * N[(0.25 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $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 3 regimes

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

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 10.2%

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

   9. Applied rewrites 10.2%

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

   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))))) < 2.0000000000000001e301

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.4%

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

   8. Applied rewrites 64.4%

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

   if 2.0000000000000001e301 < (*.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 73.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 12.9%

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

   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 13.0%

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

   7. Applied rewrites 13.0%

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

Alternative 6: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right) \cdot -2\\ t_1 := \frac{\left|U\right|}{\left|J\right|}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{0.25 \cdot {\left(\left|U\right|\right)}^{2}} \cdot t\_0\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(t\_1, 0.25 \cdot t\_1, 1\right)} \cdot \left|J\right|\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (cos (* -0.5 K)) -2.0))
        (t_1 (/ (fabs U) (fabs J)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 (fabs J)) t_2)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_2)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_3 (- INFINITY))
      (* (sqrt (* 0.25 (pow (fabs U) 2.0))) t_0)
      (if (<= t_3 2e+301)
        (* (* (sqrt (fma t_1 (* 0.25 t_1) 1.0)) (fabs J)) t_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((-0.5 * K)) * -2.0;
	double t_1 = fabs(U) / fabs(J);
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * fabs(J)) * t_2) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt((0.25 * pow(fabs(U), 2.0))) * t_0;
	} else if (t_3 <= 2e+301) {
		tmp = (sqrt(fma(t_1, (0.25 * t_1), 1.0)) * fabs(J)) * t_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 = Float64(cos(Float64(-0.5 * K)) * -2.0)
	t_1 = Float64(abs(U) / abs(J))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * abs(J)) * t_2) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(0.25 * (abs(U) ^ 2.0))) * t_0);
	elseif (t_3 <= 2e+301)
		tmp = Float64(Float64(sqrt(fma(t_1, Float64(0.25 * t_1), 1.0)) * abs(J)) * t_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[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(0.25 * N[Power[N[Abs[U], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 2e+301], N[(N[(N[Sqrt[N[(t$95$1 * N[(0.25 * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$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 3 regimes

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

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 14.7%

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

   6. Applied rewrites 14.7%

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

   7. Taylor expanded in K around 0

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

      1. lower-pow.f64 10.2%

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

   9. Applied rewrites 10.2%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e301

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.4%

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

   8. Applied rewrites 64.4%

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

   if 2.0000000000000001e301 < (*.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 73.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 12.9%

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

   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 13.0%

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

   7. Applied rewrites 13.0%

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

Alternative 7: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot \left|J\right|\right) \cdot t\_0\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t\_1 \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{\left|U\right|}{\left|J\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (* (* -2.0 (fabs J)) t_0)))
   (*
    (copysign 1.0 J)
    (if (<=
         (* t_1 (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0))))
         2e+301)
      (* t_1 (cosh (asinh (* 0.5 (/ (fabs U) (fabs J))))))
      (* 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 t_1 = (-2.0 * fabs(J)) * t_0;
	double tmp;
	if ((t_1 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)))) <= 2e+301) {
		tmp = t_1 * cosh(asinh((0.5 * (fabs(U) / fabs(J)))));
	} else {
		tmp = 2.0 * (fabs(J) * (fabs(U) * sqrt((0.25 / pow(fabs(J), 2.0)))));
	}
	return 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
	tmp = 0
	if (t_1 * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_0)), 2.0)))) <= 2e+301:
		tmp = t_1 * math.cosh(math.asinh((0.5 * (math.fabs(U) / math.fabs(J)))))
	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))
	t_1 = Float64(Float64(-2.0 * abs(J)) * t_0)
	tmp = 0.0
	if (Float64(t_1 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0)))) <= 2e+301)
		tmp = Float64(t_1 * cosh(asinh(Float64(0.5 * Float64(abs(U) / abs(J))))));
	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));
	t_1 = (-2.0 * abs(J)) * t_0;
	tmp = 0.0;
	if ((t_1 * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_0)) ^ 2.0)))) <= 2e+301)
		tmp = t_1 * cosh(asinh((0.5 * (abs(U) / abs(J)))));
	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]}, Block[{t$95$1 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+301], N[(t$95$1 * N[Cosh[N[ArcSinh[N[(0.5 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $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))))) < 2.0000000000000001e301

   1. Initial program 73.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 85.1%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 85.1%

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.1%

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.6%

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

   6. Applied rewrites 71.6%

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

   if 2.0000000000000001e301 < (*.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 73.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 12.9%

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

   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 13.0%

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

   7. Applied rewrites 13.0%

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

Alternative 8: 66.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \frac{\left|U\right|}{\left|J\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(t\_1, 0.25 \cdot t\_1, 1\right)} \cdot \left|J\right|\right) \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (/ (fabs U) (fabs J))))
   (*
    (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))))
         2e+301)
      (*
       (* (sqrt (fma t_1 (* 0.25 t_1) 1.0)) (fabs J))
       (* (cos (* -0.5 K)) -2.0))
      (* 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 t_1 = fabs(U) / fabs(J);
	double tmp;
	if ((((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)))) <= 2e+301) {
		tmp = (sqrt(fma(t_1, (0.25 * t_1), 1.0)) * fabs(J)) * (cos((-0.5 * K)) * -2.0);
	} 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))
	t_1 = Float64(abs(U) / abs(J))
	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)))) <= 2e+301)
		tmp = Float64(Float64(sqrt(fma(t_1, Float64(0.25 * t_1), 1.0)) * abs(J)) * Float64(cos(Float64(-0.5 * K)) * -2.0));
	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]}, Block[{t$95$1 = N[(N[Abs[U], $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[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], 2e+301], N[(N[(N[Sqrt[N[(t$95$1 * N[(0.25 * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $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))))) < 2.0000000000000001e301

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.4%

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

   8. Applied rewrites 64.4%

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

   if 2.0000000000000001e301 < (*.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 73.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 12.9%

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

   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 13.0%

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

   7. Applied rewrites 13.0%

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

Alternative 9: 62.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.995:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(-0.5 \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5} - -1} \cdot J\right) \cdot -2\\ \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.995)
   (* -2.0 (* J (cos (* -0.5 K))))
   (* (* (sqrt (- (/ (/ (* (/ U J) (/ U J)) 4.0) (+ 0.5 0.5)) -1.0)) J) -2.0)))

C

double code(double J, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.995) {
		tmp = -2.0 * (J * cos((-0.5 * K)));
	} else {
		tmp = (sqrt((((((U / J) * (U / J)) / 4.0) / (0.5 + 0.5)) - -1.0)) * J) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= 0.995d0) then
        tmp = (-2.0d0) * (j * cos(((-0.5d0) * k)))
    else
        tmp = (sqrt((((((u / j) * (u / j)) / 4.0d0) / (0.5d0 + 0.5d0)) - (-1.0d0))) * j) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.995) {
		tmp = -2.0 * (J * Math.cos((-0.5 * K)));
	} else {
		tmp = (Math.sqrt((((((U / J) * (U / J)) / 4.0) / (0.5 + 0.5)) - -1.0)) * J) * -2.0;
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.995:
		tmp = -2.0 * (J * math.cos((-0.5 * K)))
	else:
		tmp = (math.sqrt((((((U / J) * (U / J)) / 4.0) / (0.5 + 0.5)) - -1.0)) * J) * -2.0
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.995)
		tmp = Float64(-2.0 * Float64(J * cos(Float64(-0.5 * K))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(U / J) * Float64(U / J)) / 4.0) / Float64(0.5 + 0.5)) - -1.0)) * J) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.995)
		tmp = -2.0 * (J * cos((-0.5 * K)));
	else
		tmp = (sqrt((((((U / J) * (U / J)) / 4.0) / (0.5 + 0.5)) - -1.0)) * J) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.995], N[(-2.0 * N[(J * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[(N[(N[(U / J), $MachinePrecision] * N[(U / J), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] / N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.995

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in J around inf

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)} \]
   5. 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 52.0%

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

   6. Applied rewrites 52.0%

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

   if 0.995 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in K around 0

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

   9. Applied rewrites 44.4%

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

Alternative 10: 50.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|U\right|}{\left|J\right|}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}} \leq 2 \cdot 10^{-143}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{t\_0 \cdot t\_0}{4}}{0.5 + 0.5} - -1} \cdot \left|J\right|\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|J\right| \cdot \left(\left|U\right| \cdot \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ (fabs U) (fabs J))) (t_1 (cos (/ K 2.0))))
   (*
    (copysign 1.0 J)
    (if (<=
         (*
          (* (* -2.0 (fabs J)) t_1)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0))))
         2e-143)
      (* (* (sqrt (- (/ (/ (* t_0 t_0) 4.0) (+ 0.5 0.5)) -1.0)) (fabs J)) -2.0)
      (* 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 = fabs(U) / fabs(J);
	double t_1 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * fabs(J)) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)))) <= 2e-143) {
		tmp = (sqrt(((((t_0 * t_0) / 4.0) / (0.5 + 0.5)) - -1.0)) * fabs(J)) * -2.0;
	} 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.abs(U) / Math.abs(J);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * Math.abs(J)) * t_1) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)))) <= 2e-143) {
		tmp = (Math.sqrt(((((t_0 * t_0) / 4.0) / (0.5 + 0.5)) - -1.0)) * Math.abs(J)) * -2.0;
	} 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.fabs(U) / math.fabs(J)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * math.fabs(J)) * t_1) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))) <= 2e-143:
		tmp = (math.sqrt(((((t_0 * t_0) / 4.0) / (0.5 + 0.5)) - -1.0)) * math.fabs(J)) * -2.0
	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 = Float64(abs(U) / abs(J))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * abs(J)) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0)))) <= 2e-143)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(t_0 * t_0) / 4.0) / Float64(0.5 + 0.5)) - -1.0)) * abs(J)) * -2.0);
	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

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = abs(U) / abs(J);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * abs(J)) * t_1) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)))) <= 2e-143)
		tmp = (sqrt(((((t_0 * t_0) / 4.0) / (0.5 + 0.5)) - -1.0)) * abs(J)) * -2.0;
	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[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-143], N[(N[(N[Sqrt[N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] / N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -2.0), $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))))) < 1.9999999999999999e-143

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in K around 0

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

   9. Applied rewrites 44.4%

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

   if 1.9999999999999999e-143 < (*.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 73.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 12.9%

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

   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 13.0%

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

   7. Applied rewrites 13.0%

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

Alternative 11: 44.4% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (J K U)
 :precision binary64
 (* (* (sqrt (- (/ (/ (* (/ U J) (/ U J)) 4.0) (+ 0.5 0.5)) -1.0)) J) -2.0))

C

double code(double J, double K, double U) {
	return (sqrt((((((U / J) * (U / J)) / 4.0) / (0.5 + 0.5)) - -1.0)) * J) * -2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = (sqrt((((((u / j) * (u / j)) / 4.0d0) / (0.5d0 + 0.5d0)) - (-1.0d0))) * j) * (-2.0d0)
end function

Java

public static double code(double J, double K, double U) {
	return (Math.sqrt((((((U / J) * (U / J)) / 4.0) / (0.5 + 0.5)) - -1.0)) * J) * -2.0;
}

Python

def code(J, K, U):
	return (math.sqrt((((((U / J) * (U / J)) / 4.0) / (0.5 + 0.5)) - -1.0)) * J) * -2.0

Julia

function code(J, K, U)
	return Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(U / J) * Float64(U / J)) / 4.0) / Float64(0.5 + 0.5)) - -1.0)) * J) * -2.0)
end

MATLAB

function tmp = code(J, K, U)
	tmp = (sqrt((((((U / J) * (U / J)) / 4.0) / (0.5 + 0.5)) - -1.0)) * J) * -2.0;
end

Wolfram

code[J_, K_, U_] := N[(N[(N[Sqrt[N[(N[(N[(N[(N[(U / J), $MachinePrecision] * N[(U / J), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] / N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 73.6%

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

3. Applied rewrites 73.4%

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

4. Taylor expanded in K around 0

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

6. Applied rewrites 40.0%

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

7. Taylor expanded in K around 0

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

9. Applied rewrites 44.4%

   \[\leadsto \left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + \color{blue}{0.5}} - -1} \cdot J\right) \cdot -2 \]
10. Add Preprocessing

Alternative 12: 26.6% accurate, 6.2× 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 73.6%

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

2. Taylor expanded in J around inf

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

4. Applied rewrites 52.0%

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

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 26.6%

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

7. Applied rewrites 26.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot 1 \]
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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 26.6%

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

9. Applied rewrites 26.6%

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

Reproduce

herbie shell --seed 2025196 
(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)))))