Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 98.3%

Time: 6.2s

Alternatives: 13

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+274}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left|J\right|\right) \cdot -2\right) \cdot \sqrt{{\left(\frac{\left|U\right|}{\left(\left|J\right| + \left|J\right|\right) \cdot t\_0}\right)}^{2} - -1}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 (fabs J)) t_1)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_2 (- INFINITY))
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_2 1e+274)
        (*
         (* (* t_0 (fabs J)) -2.0)
         (sqrt (- (pow (/ (fabs U) (* (+ (fabs J) (fabs J)) t_0)) 2.0) -1.0)))
        (* -2.0 (* -0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * fabs(J)) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (0.5 * fabs(U));
	} else if (t_2 <= 1e+274) {
		tmp = ((t_0 * fabs(J)) * -2.0) * sqrt((pow((fabs(U) / ((fabs(J) + fabs(J)) * t_0)), 2.0) - -1.0));
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * Math.abs(J)) * t_1) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (0.5 * Math.abs(U));
	} else if (t_2 <= 1e+274) {
		tmp = ((t_0 * Math.abs(J)) * -2.0) * Math.sqrt((Math.pow((Math.abs(U) / ((Math.abs(J) + Math.abs(J)) * t_0)), 2.0) - -1.0));
	} else {
		tmp = -2.0 * (-0.5 * Math.abs(U));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * math.fabs(J)) * t_1) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (0.5 * math.fabs(U))
	elif t_2 <= 1e+274:
		tmp = ((t_0 * math.fabs(J)) * -2.0) * math.sqrt((math.pow((math.fabs(U) / ((math.fabs(J) + math.fabs(J)) * t_0)), 2.0) - -1.0))
	else:
		tmp = -2.0 * (-0.5 * math.fabs(U))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * abs(J)) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_2 <= 1e+274)
		tmp = Float64(Float64(Float64(t_0 * abs(J)) * -2.0) * sqrt(Float64((Float64(abs(U) / Float64(Float64(abs(J) + abs(J)) * t_0)) ^ 2.0) - -1.0)));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * abs(J)) * t_1) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (0.5 * abs(U));
	elseif (t_2 <= 1e+274)
		tmp = ((t_0 * abs(J)) * -2.0) * sqrt((((abs(U) / ((abs(J) + abs(J)) * t_0)) ^ 2.0) - -1.0));
	else
		tmp = -2.0 * (-0.5 * abs(U));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+274], N[(N[(N[(t$95$0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.99999999999999921e273

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.3

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

   3. Applied rewrites 73.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. metadata-eval 73.3

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

   7. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift-/.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lift-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. associate-*l* N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

   9. Applied rewrites 73.3%

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

   if 9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 2: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \frac{\left|U\right|}{\left|J\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+274}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{t\_2 \cdot t\_2}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ 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 (/ (fabs U) (fabs J))))
   (*
    (copysign 1.0 J)
    (if (<= t_1 (- INFINITY))
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_1 1e+274)
        (*
         (*
          (sqrt (- (/ (/ (* t_2 t_2) 4.0) (+ 0.5 (* 0.5 (cos K)))) -1.0))
          (cos (* -0.5 K)))
         (* (fabs J) -2.0))
        (* -2.0 (* -0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double t_2 = fabs(U) / fabs(J);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -2.0 * (0.5 * fabs(U));
	} else if (t_1 <= 1e+274) {
		tmp = (sqrt(((((t_2 * t_2) / 4.0) / (0.5 + (0.5 * cos(K)))) - -1.0)) * cos((-0.5 * K))) * (fabs(J) * -2.0);
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * Math.abs(J)) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_0)), 2.0)));
	double t_2 = Math.abs(U) / Math.abs(J);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (0.5 * Math.abs(U));
	} else if (t_1 <= 1e+274) {
		tmp = (Math.sqrt(((((t_2 * t_2) / 4.0) / (0.5 + (0.5 * Math.cos(K)))) - -1.0)) * Math.cos((-0.5 * K))) * (Math.abs(J) * -2.0);
	} else {
		tmp = -2.0 * (-0.5 * Math.abs(U));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * math.fabs(J)) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_0)), 2.0)))
	t_2 = math.fabs(U) / math.fabs(J)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -2.0 * (0.5 * math.fabs(U))
	elif t_1 <= 1e+274:
		tmp = (math.sqrt(((((t_2 * t_2) / 4.0) / (0.5 + (0.5 * math.cos(K)))) - -1.0)) * math.cos((-0.5 * K))) * (math.fabs(J) * -2.0)
	else:
		tmp = -2.0 * (-0.5 * math.fabs(U))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * abs(J)) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0))))
	t_2 = Float64(abs(U) / abs(J))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_1 <= 1e+274)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(t_2 * t_2) / 4.0) / Float64(0.5 + Float64(0.5 * cos(K)))) - -1.0)) * cos(Float64(-0.5 * K))) * Float64(abs(J) * -2.0));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * abs(J)) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_0)) ^ 2.0)));
	t_2 = abs(U) / abs(J);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -2.0 * (0.5 * abs(U));
	elseif (t_1 <= 1e+274)
		tmp = (sqrt(((((t_2 * t_2) / 4.0) / (0.5 + (0.5 * cos(K)))) - -1.0)) * cos((-0.5 * K))) * (abs(J) * -2.0);
	else
		tmp = -2.0 * (-0.5 * abs(U));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(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[(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[t$95$1, (-Infinity)], N[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+274], N[(N[(N[Sqrt[N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] / 4.0), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 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 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.99999999999999921e273

   1. Initial program 73.3%

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

   3. Applied rewrites 73.1%

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

   if 9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 3: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|U\right|}{\left|J\right|}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+274}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(t\_0, \frac{t\_0}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot 4}, 1\right)} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ (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))
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_2 1e+274)
        (*
         (*
          (sqrt (fma t_0 (/ t_0 (* (fma (cos K) 0.5 0.5) 4.0)) 1.0))
          (cos (* -0.5 K)))
         (* (fabs J) -2.0))
        (* -2.0 (* -0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = 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 = -2.0 * (0.5 * fabs(U));
	} else if (t_2 <= 1e+274) {
		tmp = (sqrt(fma(t_0, (t_0 / (fma(cos(K), 0.5, 0.5) * 4.0)), 1.0)) * cos((-0.5 * K))) * (fabs(J) * -2.0);
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = 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(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_2 <= 1e+274)
		tmp = Float64(Float64(sqrt(fma(t_0, Float64(t_0 / Float64(fma(cos(K), 0.5, 0.5) * 4.0)), 1.0)) * cos(Float64(-0.5 * K))) * Float64(abs(J) * -2.0));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(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[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+274], N[(N[(N[Sqrt[N[(t$95$0 * N[(t$95$0 / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 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 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.99999999999999921e273

   1. Initial program 73.3%

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

   3. Applied rewrites 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 73.1%

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

   if 9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 4: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left(\left(t\_0 \cdot -2\right) \cdot \left|J\right|\right) \cdot \sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+274}:\\ \;\;\;\;\left(\left(\left|J\right| \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\left|U\right| \cdot \frac{\left|U\right|}{\left|J\right| \cdot \left|J\right|}, \frac{-0.25}{\mathsf{fma}\left(\cos K, -0.5, -0.5\right)}, 1\right)}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (/ (fabs U) (+ (fabs J) (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))
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_3 2e-74)
        (* (* (* t_0 -2.0) (fabs J)) (sqrt (fma t_1 t_1 1.0)))
        (if (<= t_3 1e+274)
          (*
           (*
            (* (fabs J) -2.0)
            (sqrt
             (fma
              (* (fabs U) (/ (fabs U) (* (fabs J) (fabs J))))
              (/ -0.25 (fma (cos K) -0.5 -0.5))
              1.0)))
           t_0)
          (* -2.0 (* -0.5 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = fabs(U) / (fabs(J) + 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 = -2.0 * (0.5 * fabs(U));
	} else if (t_3 <= 2e-74) {
		tmp = ((t_0 * -2.0) * fabs(J)) * sqrt(fma(t_1, t_1, 1.0));
	} else if (t_3 <= 1e+274) {
		tmp = ((fabs(J) * -2.0) * sqrt(fma((fabs(U) * (fabs(U) / (fabs(J) * fabs(J)))), (-0.25 / fma(cos(K), -0.5, -0.5)), 1.0))) * t_0;
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(abs(U) / Float64(abs(J) + 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(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_3 <= 2e-74)
		tmp = Float64(Float64(Float64(t_0 * -2.0) * abs(J)) * sqrt(fma(t_1, t_1, 1.0)));
	elseif (t_3 <= 1e+274)
		tmp = Float64(Float64(Float64(abs(J) * -2.0) * sqrt(fma(Float64(abs(U) * Float64(abs(U) / Float64(abs(J) * abs(J)))), Float64(-0.25 / fma(cos(K), -0.5, -0.5)), 1.0))) * t_0);
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $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[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-74], N[(N[(N[(t$95$0 * -2.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+274], N[(N[(N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 / N[(N[Cos[K], $MachinePrecision] * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.99999999999999992e-74

   1. Initial program 73.3%

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

      1. lift-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 85.0

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

      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 85.0

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

      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 85.0

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

   3. Applied rewrites 85.0%

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

   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 71.5

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

   6. Applied rewrites 71.5%

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

   8. Applied rewrites 64.6%

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

   if 1.99999999999999992e-74 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.99999999999999921e273

   1. Initial program 73.3%

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 61.1%

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

   if 9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 5: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\left(\left(t\_0 \cdot -2\right) \cdot \left|J\right|\right) \cdot \sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+274}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{\frac{\left|U\right|}{\left|J\right| \cdot \left|J\right|} \cdot \left|U\right|}{\cos K - -1}, 0.5, 1\right)} \cdot t\_0\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 K)))
        (t_1 (/ (fabs U) (+ (fabs J) (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))
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_3 2e-74)
        (* (* (* t_0 -2.0) (fabs J)) (sqrt (fma t_1 t_1 1.0)))
        (if (<= t_3 1e+274)
          (*
           (*
            (sqrt
             (fma
              (/
               (* (/ (fabs U) (* (fabs J) (fabs J))) (fabs U))
               (- (cos K) -1.0))
              0.5
              1.0))
            t_0)
           (* (fabs J) -2.0))
          (* -2.0 (* -0.5 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = fabs(U) / (fabs(J) + 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 = -2.0 * (0.5 * fabs(U));
	} else if (t_3 <= 2e-74) {
		tmp = ((t_0 * -2.0) * fabs(J)) * sqrt(fma(t_1, t_1, 1.0));
	} else if (t_3 <= 1e+274) {
		tmp = (sqrt(fma((((fabs(U) / (fabs(J) * fabs(J))) * fabs(U)) / (cos(K) - -1.0)), 0.5, 1.0)) * t_0) * (fabs(J) * -2.0);
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(abs(U) / Float64(abs(J) + 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(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_3 <= 2e-74)
		tmp = Float64(Float64(Float64(t_0 * -2.0) * abs(J)) * sqrt(fma(t_1, t_1, 1.0)));
	elseif (t_3 <= 1e+274)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(Float64(abs(U) / Float64(abs(J) * abs(J))) * abs(U)) / Float64(cos(K) - -1.0)), 0.5, 1.0)) * t_0) * Float64(abs(J) * -2.0));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $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[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-74], N[(N[(N[(t$95$0 * -2.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+274], N[(N[(N[Sqrt[N[(N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[K], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.99999999999999992e-74

   1. Initial program 73.3%

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

      1. lift-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 85.0

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

      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 85.0

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

      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 85.0

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

   3. Applied rewrites 85.0%

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

   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 71.5

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

   6. Applied rewrites 71.5%

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

   8. Applied rewrites 64.6%

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

   if 1.99999999999999992e-74 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.99999999999999921e273

   1. Initial program 73.3%

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 61.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval 61.1

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

   7. Applied rewrites 61.1%

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

   if 9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 6: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|U\right|}{\left|J\right| + \left|J\right|}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+274}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot \left|J\right|\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ (fabs U) (+ (fabs J) (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))
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_2 1e+274)
        (* (* (* (cos (* -0.5 K)) -2.0) (fabs J)) (sqrt (fma t_0 t_0 1.0)))
        (* -2.0 (* -0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(U) / (fabs(J) + fabs(J));
	double t_1 = cos((K / 2.0));
	double 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 = -2.0 * (0.5 * fabs(U));
	} else if (t_2 <= 1e+274) {
		tmp = ((cos((-0.5 * K)) * -2.0) * fabs(J)) * sqrt(fma(t_0, t_0, 1.0));
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(abs(U) / Float64(abs(J) + abs(J)))
	t_1 = cos(Float64(K / 2.0))
	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(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_2 <= 1e+274)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * -2.0) * abs(J)) * sqrt(fma(t_0, t_0, 1.0)));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, 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[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+274], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.99999999999999921e273

   1. Initial program 73.3%

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

      1. lift-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 85.0

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

      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 85.0

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

      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 85.0

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

   3. Applied rewrites 85.0%

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

   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 71.5

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

   6. Applied rewrites 71.5%

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

   8. Applied rewrites 64.6%

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

   if 9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 7: 83.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|J\right| \cdot -2\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \cos \left(-0.5 \cdot K\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+297}:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\left|U\right| \cdot \frac{\left|U\right|}{\left|J\right| \cdot \left|J\right|}, 0.25, 1\right)} \cdot t\_3\right) \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq 10^{+274}:\\ \;\;\;\;t\_3 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (fabs J) -2.0))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 (fabs J)) t_1)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0)))))
        (t_3 (cos (* -0.5 K))))
   (*
    (copysign 1.0 J)
    (if (<= t_2 -1e+297)
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_2 -1e-102)
        (*
         (*
          (sqrt (fma (* (fabs U) (/ (fabs U) (* (fabs J) (fabs J)))) 0.25 1.0))
          t_3)
         t_0)
        (if (<= t_2 1e+274) (* t_3 t_0) (* -2.0 (* -0.5 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(J) * -2.0;
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * fabs(J)) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double t_3 = cos((-0.5 * K));
	double tmp;
	if (t_2 <= -1e+297) {
		tmp = -2.0 * (0.5 * fabs(U));
	} else if (t_2 <= -1e-102) {
		tmp = (sqrt(fma((fabs(U) * (fabs(U) / (fabs(J) * fabs(J)))), 0.25, 1.0)) * t_3) * t_0;
	} else if (t_2 <= 1e+274) {
		tmp = t_3 * t_0;
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(abs(J) * -2.0)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * abs(J)) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	t_3 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (t_2 <= -1e+297)
		tmp = Float64(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_2 <= -1e-102)
		tmp = Float64(Float64(sqrt(fma(Float64(abs(U) * Float64(abs(U) / Float64(abs(J) * abs(J)))), 0.25, 1.0)) * t_3) * t_0);
	elseif (t_2 <= 1e+274)
		tmp = Float64(t_3 * t_0);
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1e+297], N[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-102], N[(N[(N[Sqrt[N[(N[(N[Abs[U], $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 1e+274], N[(t$95$3 * t$95$0), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e297

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -1e297 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999933e-103

   1. Initial program 73.3%

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 56.5%

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

   if -9.99999999999999933e-103 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.99999999999999921e273

   1. Initial program 73.3%

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

   3. Applied rewrites 73.1%

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

   4. Taylor expanded in J around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if 9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 8: 78.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right) \cdot \left(\left|J\right| \cdot -2\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+297}:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;-2 \cdot \left(\left|J\right| \cdot \sqrt{1 + 0.25 \cdot \frac{{\left(\left|U\right|\right)}^{2}}{{\left(\left|J\right|\right)}^{2}}}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+274}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (cos (* -0.5 K)) (* (fabs J) -2.0)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 (fabs J)) t_1)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_2 -1e+297)
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_2 -5e+160)
        t_0
        (if (<= t_2 -1e-102)
          (*
           -2.0
           (*
            (fabs J)
            (sqrt (+ 1.0 (* 0.25 (/ (pow (fabs U) 2.0) (pow (fabs J) 2.0)))))))
          (if (<= t_2 1e+274) t_0 (* -2.0 (* -0.5 (fabs U))))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K)) * (fabs(J) * -2.0);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * fabs(J)) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -1e+297) {
		tmp = -2.0 * (0.5 * fabs(U));
	} else if (t_2 <= -5e+160) {
		tmp = t_0;
	} else if (t_2 <= -1e-102) {
		tmp = -2.0 * (fabs(J) * sqrt((1.0 + (0.25 * (pow(fabs(U), 2.0) / pow(fabs(J), 2.0))))));
	} else if (t_2 <= 1e+274) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K)) * (Math.abs(J) * -2.0);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * Math.abs(J)) * t_1) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -1e+297) {
		tmp = -2.0 * (0.5 * Math.abs(U));
	} else if (t_2 <= -5e+160) {
		tmp = t_0;
	} else if (t_2 <= -1e-102) {
		tmp = -2.0 * (Math.abs(J) * Math.sqrt((1.0 + (0.25 * (Math.pow(Math.abs(U), 2.0) / Math.pow(Math.abs(J), 2.0))))));
	} else if (t_2 <= 1e+274) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (-0.5 * Math.abs(U));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K)) * (math.fabs(J) * -2.0)
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * math.fabs(J)) * t_1) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -1e+297:
		tmp = -2.0 * (0.5 * math.fabs(U))
	elif t_2 <= -5e+160:
		tmp = t_0
	elif t_2 <= -1e-102:
		tmp = -2.0 * (math.fabs(J) * math.sqrt((1.0 + (0.25 * (math.pow(math.fabs(U), 2.0) / math.pow(math.fabs(J), 2.0))))))
	elif t_2 <= 1e+274:
		tmp = t_0
	else:
		tmp = -2.0 * (-0.5 * math.fabs(U))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(cos(Float64(-0.5 * K)) * Float64(abs(J) * -2.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * abs(J)) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -1e+297)
		tmp = Float64(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_2 <= -5e+160)
		tmp = t_0;
	elseif (t_2 <= -1e-102)
		tmp = Float64(-2.0 * Float64(abs(J) * sqrt(Float64(1.0 + Float64(0.25 * Float64((abs(U) ^ 2.0) / (abs(J) ^ 2.0)))))));
	elseif (t_2 <= 1e+274)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K)) * (abs(J) * -2.0);
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * abs(J)) * t_1) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -1e+297)
		tmp = -2.0 * (0.5 * abs(U));
	elseif (t_2 <= -5e+160)
		tmp = t_0;
	elseif (t_2 <= -1e-102)
		tmp = -2.0 * (abs(J) * sqrt((1.0 + (0.25 * ((abs(U) ^ 2.0) / (abs(J) ^ 2.0))))));
	elseif (t_2 <= 1e+274)
		tmp = t_0;
	else
		tmp = -2.0 * (-0.5 * abs(U));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $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, -1e+297], N[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e+160], t$95$0, If[LessEqual[t$95$2, -1e-102], N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[N[Abs[U], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Abs[J], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+274], t$95$0, N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e297

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -1e297 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e160 or -9.99999999999999933e-103 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.99999999999999921e273

   1. Initial program 73.3%

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

   3. Applied rewrites 73.1%

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

   4. Taylor expanded in J around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if -5.0000000000000002e160 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999933e-103

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 32.3

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

   4. Applied rewrites 32.3%

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

   if 9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 9: 77.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+297}:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+274}:\\ \;\;\;\;\cos \left(-0.5 \cdot K\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ 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))))))
   (*
    (copysign 1.0 J)
    (if (<= t_1 -1e+297)
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_1 1e+274)
        (* (cos (* -0.5 K)) (* (fabs J) -2.0))
        (* -2.0 (* -0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -1e+297) {
		tmp = -2.0 * (0.5 * fabs(U));
	} else if (t_1 <= 1e+274) {
		tmp = cos((-0.5 * K)) * (fabs(J) * -2.0);
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * Math.abs(J)) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -1e+297) {
		tmp = -2.0 * (0.5 * Math.abs(U));
	} else if (t_1 <= 1e+274) {
		tmp = Math.cos((-0.5 * K)) * (Math.abs(J) * -2.0);
	} else {
		tmp = -2.0 * (-0.5 * Math.abs(U));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * math.fabs(J)) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -1e+297:
		tmp = -2.0 * (0.5 * math.fabs(U))
	elif t_1 <= 1e+274:
		tmp = math.cos((-0.5 * K)) * (math.fabs(J) * -2.0)
	else:
		tmp = -2.0 * (-0.5 * math.fabs(U))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(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))))
	tmp = 0.0
	if (t_1 <= -1e+297)
		tmp = Float64(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_1 <= 1e+274)
		tmp = Float64(cos(Float64(-0.5 * K)) * Float64(abs(J) * -2.0));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * abs(J)) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -1e+297)
		tmp = -2.0 * (0.5 * abs(U));
	elseif (t_1 <= 1e+274)
		tmp = cos((-0.5 * K)) * (abs(J) * -2.0);
	else
		tmp = -2.0 * (-0.5 * abs(U));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -1e+297], N[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+274], N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 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))))) < -1e297

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -1e297 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.99999999999999921e273

   1. Initial program 73.3%

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

   3. Applied rewrites 73.1%

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

   4. Taylor expanded in J around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if 9.99999999999999921e273 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 10: 59.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+280}:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(K \cdot K\right) \cdot \left|J\right|, -0.005208333333333333, 0.25 \cdot \left|J\right|\right) \cdot K, K, \left|J\right| \cdot -2\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 (fabs J)) t_0)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_1 -1e+280)
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_1 -5e-285)
        (*
         (fma
          (*
           (fma (* (* K K) (fabs J)) -0.005208333333333333 (* 0.25 (fabs J)))
           K)
          K
          (* (fabs J) -2.0))
         1.0)
        (* -2.0 (* -0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -1e+280) {
		tmp = -2.0 * (0.5 * fabs(U));
	} else if (t_1 <= -5e-285) {
		tmp = fma((fma(((K * K) * fabs(J)), -0.005208333333333333, (0.25 * fabs(J))) * K), K, (fabs(J) * -2.0)) * 1.0;
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * abs(J)) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1e+280)
		tmp = Float64(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_1 <= -5e-285)
		tmp = Float64(fma(Float64(fma(Float64(Float64(K * K) * abs(J)), -0.005208333333333333, Float64(0.25 * abs(J))) * K), K, Float64(abs(J) * -2.0)) * 1.0);
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -1e+280], N[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-285], N[(N[(N[(N[(N[(N[(K * K), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -0.005208333333333333 + N[(0.25 * N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * K), $MachinePrecision] * K + N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e280

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -1e280 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000018e-285

   1. Initial program 73.3%

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

   2. Taylor expanded in J around inf

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 27.5

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

   7. Applied rewrites 27.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 27.5%

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

   if -5.00000000000000018e-285 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 11: 58.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+279}:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.25 \cdot \left|J\right|\right) \cdot K, K, \left|J\right| \cdot -2\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 (fabs J)) t_0)
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_0)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_1 -5e+279)
      (* -2.0 (* 0.5 (fabs U)))
      (if (<= t_1 -5e-285)
        (* (fma (* (* 0.25 (fabs J)) K) K (* (fabs J) -2.0)) 1.0)
        (* -2.0 (* -0.5 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * fabs(J)) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -5e+279) {
		tmp = -2.0 * (0.5 * fabs(U));
	} else if (t_1 <= -5e-285) {
		tmp = fma(((0.25 * fabs(J)) * K), K, (fabs(J) * -2.0)) * 1.0;
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * abs(J)) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+279)
		tmp = Float64(-2.0 * Float64(0.5 * abs(U)));
	elseif (t_1 <= -5e-285)
		tmp = Float64(fma(Float64(Float64(0.25 * abs(J)) * K), K, Float64(abs(J) * -2.0)) * 1.0);
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -5e+279], N[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-285], N[(N[(N[(N[(0.25 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * K), $MachinePrecision] * K + N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e279

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -5.0000000000000002e279 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000018e-285

   1. Initial program 73.3%

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

   2. Taylor expanded in J around inf

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 27.7

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

   7. Applied rewrites 27.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      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. lower-*.f64 27.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 27.7

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

   9. Applied rewrites 27.7%

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

   if -5.00000000000000018e-285 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 12: 51.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot \left|U\right|\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \left|U\right|\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<=
        (*
         (* (* -2.0 J) t_0)
         (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 J) t_0)) 2.0))))
        -5e-285)
     (* -2.0 (* 0.5 (fabs U)))
     (* -2.0 (* -0.5 (fabs U))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_0)), 2.0)))) <= -5e-285) {
		tmp = -2.0 * (0.5 * fabs(U));
	} else {
		tmp = -2.0 * (-0.5 * fabs(U));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((abs(u) / ((2.0d0 * j) * t_0)) ** 2.0d0)))) <= (-5d-285)) then
        tmp = (-2.0d0) * (0.5d0 * abs(u))
    else
        tmp = (-2.0d0) * ((-0.5d0) * abs(u))
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * J) * t_0)), 2.0)))) <= -5e-285) {
		tmp = -2.0 * (0.5 * Math.abs(U));
	} else {
		tmp = -2.0 * (-0.5 * Math.abs(U));
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * J) * t_0)), 2.0)))) <= -5e-285:
		tmp = -2.0 * (0.5 * math.fabs(U))
	else:
		tmp = -2.0 * (-0.5 * math.fabs(U))
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_0)) ^ 2.0)))) <= -5e-285)
		tmp = Float64(-2.0 * Float64(0.5 * abs(U)));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * abs(U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * J) * t_0)) ^ 2.0)))) <= -5e-285)
		tmp = -2.0 * (0.5 * abs(U));
	else
		tmp = -2.0 * (-0.5 * abs(U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-285], N[(-2.0 * N[(0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000018e-285

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 26.8

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

   10. Applied rewrites 26.8%

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

   if -5.00000000000000018e-285 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 15.5

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 15.2

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 15.2%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 26.3

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

   10. Applied rewrites 26.3%

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

Alternative 13: 26.6% accurate, 8.6× speedup

Math

\[\mathsf{copysign}\left(1, J\right) \cdot \left(-2 \cdot \left(-0.5 \cdot U\right)\right) \]

FPCore

(FPCore (J K U) :precision binary64 (* (copysign 1.0 J) (* -2.0 (* -0.5 U))))

C

double code(double J, double K, double U) {
	return copysign(1.0, J) * (-2.0 * (-0.5 * U));
}

Java

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

Python

def code(J, K, U):
	return math.copysign(1.0, J) * (-2.0 * (-0.5 * U))

Julia

function code(J, K, U)
	return Float64(copysign(1.0, J) * Float64(-2.0 * Float64(-0.5 * U)))
end

MATLAB

function tmp = code(J, K, U)
	tmp = (sign(J) * abs(1.0)) * (-2.0 * (-0.5 * U));
end

Wolfram

code[J_, K_, U_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(-2.0 * N[(-0.5 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in J around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-*.f64 15.5

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

4. Applied rewrites 15.5%

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

5. Taylor expanded in K around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 15.2

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

7. Applied rewrites 15.2%

   \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

8. Taylor expanded in U around -inf

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

   1. lower-*.f64 26.3

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

10. Applied rewrites 26.3%

    \[\leadsto -2 \cdot \left(-0.5 \cdot U\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025180 
(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)))))