Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.6% → 99.3%

Time: 8.1s

Alternatives: 11

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))))
  (* (* (* -2 J) t_0) (sqrt (+ 1 (pow (/ U (* (* 2 J) t_0)) 2))))))

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), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(U / N[(N[(2 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 72.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2))))
  (* (* (* -2 J) t_0) (sqrt (+ 1 (pow (/ U (* (* 2 J) t_0)) 2))))))

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), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(U / N[(N[(2 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{-1}{2} \cdot K\right)\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := 2 \cdot \left|J\right|\\ t_4 := \left(t\_1 \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{t\_3 \cdot t\_2}\right)}^{2}}\\ t_5 := \cos \left(K \cdot \frac{1}{2}\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \frac{t\_0}{\left|t\_0\right|}\right) \cdot \left|U\right|\\ \mathbf{elif}\;t\_4 \leq 4999999999999999906743388603115020788907780359910290665049241860223423941639750419942148863391427290368681348502011290786385146843522467955007764480084024749443603611970102342099448132228169829243943975742290002451379260550207232245491981306595417943121645130212363962285255265070690291922501632:\\ \;\;\;\;\left(t\_1 \cdot t\_5\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{t\_3 \cdot t\_5}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -1/2 K)))
       (t_1 (* -2 (fabs J)))
       (t_2 (cos (/ K 2)))
       (t_3 (* 2 (fabs J)))
       (t_4
        (* (* t_1 t_2) (sqrt (+ 1 (pow (/ (fabs U) (* t_3 t_2)) 2)))))
       (t_5 (cos (* K 1/2))))
  (*
   (copysign 1 J)
   (if (<= t_4 (- INFINITY))
     (* (* -1 (/ t_0 (fabs t_0))) (fabs U))
     (if (<=
          t_4
          4999999999999999906743388603115020788907780359910290665049241860223423941639750419942148863391427290368681348502011290786385146843522467955007764480084024749443603611970102342099448132228169829243943975742290002451379260550207232245491981306595417943121645130212363962285255265070690291922501632)
       (* (* t_1 t_5) (sqrt (+ 1 (pow (/ (fabs U) (* t_3 t_5)) 2))))
       (* 2 (* 1/2 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = -2.0 * fabs(J);
	double t_2 = cos((K / 2.0));
	double t_3 = 2.0 * fabs(J);
	double t_4 = (t_1 * t_2) * sqrt((1.0 + pow((fabs(U) / (t_3 * t_2)), 2.0)));
	double t_5 = cos((K * 0.5));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (-1.0 * (t_0 / fabs(t_0))) * fabs(U);
	} else if (t_4 <= 5e+294) {
		tmp = (t_1 * t_5) * sqrt((1.0 + pow((fabs(U) / (t_3 * t_5)), 2.0)));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = -2.0 * Math.abs(J);
	double t_2 = Math.cos((K / 2.0));
	double t_3 = 2.0 * Math.abs(J);
	double t_4 = (t_1 * t_2) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / (t_3 * t_2)), 2.0)));
	double t_5 = Math.cos((K * 0.5));
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = (-1.0 * (t_0 / Math.abs(t_0))) * Math.abs(U);
	} else if (t_4 <= 5e+294) {
		tmp = (t_1 * t_5) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / (t_3 * t_5)), 2.0)));
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	t_1 = -2.0 * math.fabs(J)
	t_2 = math.cos((K / 2.0))
	t_3 = 2.0 * math.fabs(J)
	t_4 = (t_1 * t_2) * math.sqrt((1.0 + math.pow((math.fabs(U) / (t_3 * t_2)), 2.0)))
	t_5 = math.cos((K * 0.5))
	tmp = 0
	if t_4 <= -math.inf:
		tmp = (-1.0 * (t_0 / math.fabs(t_0))) * math.fabs(U)
	elif t_4 <= 5e+294:
		tmp = (t_1 * t_5) * math.sqrt((1.0 + math.pow((math.fabs(U) / (t_3 * t_5)), 2.0)))
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = Float64(-2.0 * abs(J))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(2.0 * abs(J))
	t_4 = Float64(Float64(t_1 * t_2) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(t_3 * t_2)) ^ 2.0))))
	t_5 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(t_0 / abs(t_0))) * abs(U));
	elseif (t_4 <= 5e+294)
		tmp = Float64(Float64(t_1 * t_5) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(t_3 * t_5)) ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K));
	t_1 = -2.0 * abs(J);
	t_2 = cos((K / 2.0));
	t_3 = 2.0 * abs(J);
	t_4 = (t_1 * t_2) * sqrt((1.0 + ((abs(U) / (t_3 * t_2)) ^ 2.0)));
	t_5 = cos((K * 0.5));
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = (-1.0 * (t_0 / abs(t_0))) * abs(U);
	elseif (t_4 <= 5e+294)
		tmp = (t_1 * t_5) * sqrt((1.0 + ((abs(U) / (t_3 * t_5)) ^ 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[(-1/2 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(2 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(K * 1/2), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], N[(N[(-1 * N[(t$95$0 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4999999999999999906743388603115020788907780359910290665049241860223423941639750419942148863391427290368681348502011290786385146843522467955007764480084024749443603611970102342099448132228169829243943975742290002451379260550207232245491981306595417943121645130212363962285255265070690291922501632], N[(N[(t$95$1 * t$95$5), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(t$95$3 * t$95$5), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2 * N[(1/2 * 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 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 15.1%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 26.7%

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

   9. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999999e294

   1. Initial program 72.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 72.6%

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

   3. Applied rewrites 72.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 72.6%

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

   5. Applied rewrites 72.6%

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

   if 4.9999999999999999e294 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{-1}{2} \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \left|J\right| + \left|J\right|\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \frac{t\_0}{\left|t\_0\right|}\right) \cdot \left|U\right|\\ \mathbf{elif}\;t\_2 \leq 4999999999999999906743388603115020788907780359910290665049241860223423941639750419942148863391427290368681348502011290786385146843522467955007764480084024749443603611970102342099448132228169829243943975742290002451379260550207232245491981306595417943121645130212363962285255265070690291922501632:\\ \;\;\;\;\left(t\_3 \cdot \left(-t\_0\right)\right) \cdot \sqrt{1 + \frac{\left|U\right|}{t\_3} \cdot \frac{\left|U\right|}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right) \cdot t\_3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -1/2 K)))
       (t_1 (cos (/ K 2)))
       (t_2
        (*
         (* (* -2 (fabs J)) t_1)
         (sqrt (+ 1 (pow (/ (fabs U) (* (* 2 (fabs J)) t_1)) 2)))))
       (t_3 (+ (fabs J) (fabs J))))
  (*
   (copysign 1 J)
   (if (<= t_2 (- INFINITY))
     (* (* -1 (/ t_0 (fabs t_0))) (fabs U))
     (if (<=
          t_2
          4999999999999999906743388603115020788907780359910290665049241860223423941639750419942148863391427290368681348502011290786385146843522467955007764480084024749443603611970102342099448132228169829243943975742290002451379260550207232245491981306595417943121645130212363962285255265070690291922501632)
       (*
        (* t_3 (- t_0))
        (sqrt
         (+
          1
          (*
           (/ (fabs U) t_3)
           (/ (fabs U) (* (+ 1/2 (* 1/2 (cos K))) t_3))))))
       (* 2 (* 1/2 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * fabs(J)) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double t_3 = fabs(J) + fabs(J);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (-1.0 * (t_0 / fabs(t_0))) * fabs(U);
	} else if (t_2 <= 5e+294) {
		tmp = (t_3 * -t_0) * sqrt((1.0 + ((fabs(U) / t_3) * (fabs(U) / ((0.5 + (0.5 * cos(K))) * t_3)))));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * Math.abs(J)) * t_1) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double t_3 = Math.abs(J) + Math.abs(J);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (-1.0 * (t_0 / Math.abs(t_0))) * Math.abs(U);
	} else if (t_2 <= 5e+294) {
		tmp = (t_3 * -t_0) * Math.sqrt((1.0 + ((Math.abs(U) / t_3) * (Math.abs(U) / ((0.5 + (0.5 * Math.cos(K))) * t_3)))));
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * math.fabs(J)) * t_1) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	t_3 = math.fabs(J) + math.fabs(J)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (-1.0 * (t_0 / math.fabs(t_0))) * math.fabs(U)
	elif t_2 <= 5e+294:
		tmp = (t_3 * -t_0) * math.sqrt((1.0 + ((math.fabs(U) / t_3) * (math.fabs(U) / ((0.5 + (0.5 * math.cos(K))) * t_3)))))
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * abs(J)) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	t_3 = Float64(abs(J) + abs(J))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(t_0 / abs(t_0))) * abs(U));
	elseif (t_2 <= 5e+294)
		tmp = Float64(Float64(t_3 * Float64(-t_0)) * sqrt(Float64(1.0 + Float64(Float64(abs(U) / t_3) * Float64(abs(U) / Float64(Float64(0.5 + Float64(0.5 * cos(K))) * t_3))))));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * abs(J)) * t_1) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	t_3 = abs(J) + abs(J);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (-1.0 * (t_0 / abs(t_0))) * abs(U);
	elseif (t_2 <= 5e+294)
		tmp = (t_3 * -t_0) * sqrt((1.0 + ((abs(U) / t_3) * (abs(U) / ((0.5 + (0.5 * cos(K))) * t_3)))));
	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[(-1/2 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(N[(-1 * N[(t$95$0 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4999999999999999906743388603115020788907780359910290665049241860223423941639750419942148863391427290368681348502011290786385146843522467955007764480084024749443603611970102342099448132228169829243943975742290002451379260550207232245491981306595417943121645130212363962285255265070690291922501632], N[(N[(t$95$3 * (-t$95$0)), $MachinePrecision] * N[Sqrt[N[(1 + N[(N[(N[Abs[U], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[(1/2 + N[(1/2 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2 * N[(1/2 * 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 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 15.1%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 26.7%

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

   9. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999999e294

   1. Initial program 72.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 72.5%

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 72.5%

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

   5. Applied rewrites 72.5%

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

   if 4.9999999999999999e294 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{-1}{2} \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \left|J\right| + \left|J\right|\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \frac{t\_0}{\left|t\_0\right|}\right) \cdot \left|U\right|\\ \mathbf{elif}\;t\_2 \leq 4999999999999999906743388603115020788907780359910290665049241860223423941639750419942148863391427290368681348502011290786385146843522467955007764480084024749443603611970102342099448132228169829243943975742290002451379260550207232245491981306595417943121645130212363962285255265070690291922501632:\\ \;\;\;\;\left(t\_3 \cdot \left(-t\_0\right)\right) \cdot \sqrt{1 + \left|U\right| \cdot \frac{\frac{\left|U\right|}{\left(\cos K \cdot \frac{1}{2} - \frac{-1}{2}\right) \cdot t\_3}}{t\_3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -1/2 K)))
       (t_1 (cos (/ K 2)))
       (t_2
        (*
         (* (* -2 (fabs J)) t_1)
         (sqrt (+ 1 (pow (/ (fabs U) (* (* 2 (fabs J)) t_1)) 2)))))
       (t_3 (+ (fabs J) (fabs J))))
  (*
   (copysign 1 J)
   (if (<= t_2 (- INFINITY))
     (* (* -1 (/ t_0 (fabs t_0))) (fabs U))
     (if (<=
          t_2
          4999999999999999906743388603115020788907780359910290665049241860223423941639750419942148863391427290368681348502011290786385146843522467955007764480084024749443603611970102342099448132228169829243943975742290002451379260550207232245491981306595417943121645130212363962285255265070690291922501632)
       (*
        (* t_3 (- t_0))
        (sqrt
         (+
          1
          (*
           (fabs U)
           (/ (/ (fabs U) (* (- (* (cos K) 1/2) -1/2) t_3)) t_3)))))
       (* 2 (* 1/2 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * fabs(J)) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double t_3 = fabs(J) + fabs(J);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (-1.0 * (t_0 / fabs(t_0))) * fabs(U);
	} else if (t_2 <= 5e+294) {
		tmp = (t_3 * -t_0) * sqrt((1.0 + (fabs(U) * ((fabs(U) / (((cos(K) * 0.5) - -0.5) * t_3)) / t_3))));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * Math.abs(J)) * t_1) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double t_3 = Math.abs(J) + Math.abs(J);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (-1.0 * (t_0 / Math.abs(t_0))) * Math.abs(U);
	} else if (t_2 <= 5e+294) {
		tmp = (t_3 * -t_0) * Math.sqrt((1.0 + (Math.abs(U) * ((Math.abs(U) / (((Math.cos(K) * 0.5) - -0.5) * t_3)) / t_3))));
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * math.fabs(J)) * t_1) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	t_3 = math.fabs(J) + math.fabs(J)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (-1.0 * (t_0 / math.fabs(t_0))) * math.fabs(U)
	elif t_2 <= 5e+294:
		tmp = (t_3 * -t_0) * math.sqrt((1.0 + (math.fabs(U) * ((math.fabs(U) / (((math.cos(K) * 0.5) - -0.5) * t_3)) / t_3))))
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * abs(J)) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	t_3 = Float64(abs(J) + abs(J))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(t_0 / abs(t_0))) * abs(U));
	elseif (t_2 <= 5e+294)
		tmp = Float64(Float64(t_3 * Float64(-t_0)) * sqrt(Float64(1.0 + Float64(abs(U) * Float64(Float64(abs(U) / Float64(Float64(Float64(cos(K) * 0.5) - -0.5) * t_3)) / t_3)))));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * abs(J)) * t_1) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	t_3 = abs(J) + abs(J);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (-1.0 * (t_0 / abs(t_0))) * abs(U);
	elseif (t_2 <= 5e+294)
		tmp = (t_3 * -t_0) * sqrt((1.0 + (abs(U) * ((abs(U) / (((cos(K) * 0.5) - -0.5) * t_3)) / t_3))));
	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[(-1/2 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(N[(-1 * N[(t$95$0 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4999999999999999906743388603115020788907780359910290665049241860223423941639750419942148863391427290368681348502011290786385146843522467955007764480084024749443603611970102342099448132228169829243943975742290002451379260550207232245491981306595417943121645130212363962285255265070690291922501632], N[(N[(t$95$3 * (-t$95$0)), $MachinePrecision] * N[Sqrt[N[(1 + N[(N[Abs[U], $MachinePrecision] * N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(N[(N[Cos[K], $MachinePrecision] * 1/2), $MachinePrecision] - -1/2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2 * N[(1/2 * 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 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 15.1%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 26.7%

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

   9. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999999e294

   1. Initial program 72.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 72.5%

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 72.5%

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

   5. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 69.7%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 69.7%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 69.7%

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

   7. Applied rewrites 69.7%

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

   if 4.9999999999999999e294 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{-1}{2} \cdot K\right)\\ t_1 := -t\_0\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \left|J\right| + \left|J\right|\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \frac{t\_0}{\left|t\_0\right|}\right) \cdot \left|U\right|\\ \mathbf{elif}\;t\_3 \leq \frac{4717453031026927}{943490606205385338060388645247067222729230305104110107094051575061406040598037213021531681294414691885367093757690961224942646157481198158140358562858174010912348831744}:\\ \;\;\;\;\left(t\_4 \cdot t\_1\right) \cdot \sqrt{1 + \frac{\left|U\right|}{t\_4} \cdot \frac{\left|U\right|}{\left(\frac{1}{2} + \frac{1}{2}\right) \cdot t\_4}}\\ \mathbf{elif}\;t\_3 \leq 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880:\\ \;\;\;\;t\_4 \cdot \left(t\_1 \cdot \sqrt{\frac{\left|U\right|}{\left(t\_4 \cdot t\_4\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} \cdot \left|U\right| - -1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -1/2 K)))
       (t_1 (- t_0))
       (t_2 (cos (/ K 2)))
       (t_3
        (*
         (* (* -2 (fabs J)) t_2)
         (sqrt (+ 1 (pow (/ (fabs U) (* (* 2 (fabs J)) t_2)) 2)))))
       (t_4 (+ (fabs J) (fabs J))))
  (*
   (copysign 1 J)
   (if (<= t_3 (- INFINITY))
     (* (* -1 (/ t_0 (fabs t_0))) (fabs U))
     (if (<=
          t_3
          4717453031026927/943490606205385338060388645247067222729230305104110107094051575061406040598037213021531681294414691885367093757690961224942646157481198158140358562858174010912348831744)
       (*
        (* t_4 t_1)
        (sqrt
         (+ 1 (* (/ (fabs U) t_4) (/ (fabs U) (* (+ 1/2 1/2) t_4))))))
       (if (<=
            t_3
            500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880)
         (*
          t_4
          (*
           t_1
           (sqrt
            (-
             (*
              (/ (fabs U) (* (* t_4 t_4) (+ 1/2 (* 1/2 (cos K)))))
              (fabs U))
             -1))))
         (* 2 (* 1/2 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = -t_0;
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * fabs(J)) * t_2) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_4 = fabs(J) + fabs(J);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (-1.0 * (t_0 / fabs(t_0))) * fabs(U);
	} else if (t_3 <= 5e-153) {
		tmp = (t_4 * t_1) * sqrt((1.0 + ((fabs(U) / t_4) * (fabs(U) / ((0.5 + 0.5) * t_4)))));
	} else if (t_3 <= 5e+269) {
		tmp = t_4 * (t_1 * sqrt((((fabs(U) / ((t_4 * t_4) * (0.5 + (0.5 * cos(K))))) * fabs(U)) - -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 = -t_0;
	double t_2 = Math.cos((K / 2.0));
	double t_3 = ((-2.0 * Math.abs(J)) * t_2) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_2)), 2.0)));
	double t_4 = Math.abs(J) + Math.abs(J);
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (-1.0 * (t_0 / Math.abs(t_0))) * Math.abs(U);
	} else if (t_3 <= 5e-153) {
		tmp = (t_4 * t_1) * Math.sqrt((1.0 + ((Math.abs(U) / t_4) * (Math.abs(U) / ((0.5 + 0.5) * t_4)))));
	} else if (t_3 <= 5e+269) {
		tmp = t_4 * (t_1 * Math.sqrt((((Math.abs(U) / ((t_4 * t_4) * (0.5 + (0.5 * Math.cos(K))))) * Math.abs(U)) - -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 = -t_0
	t_2 = math.cos((K / 2.0))
	t_3 = ((-2.0 * math.fabs(J)) * t_2) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_2)), 2.0)))
	t_4 = math.fabs(J) + math.fabs(J)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (-1.0 * (t_0 / math.fabs(t_0))) * math.fabs(U)
	elif t_3 <= 5e-153:
		tmp = (t_4 * t_1) * math.sqrt((1.0 + ((math.fabs(U) / t_4) * (math.fabs(U) / ((0.5 + 0.5) * t_4)))))
	elif t_3 <= 5e+269:
		tmp = t_4 * (t_1 * math.sqrt((((math.fabs(U) / ((t_4 * t_4) * (0.5 + (0.5 * math.cos(K))))) * math.fabs(U)) - -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 = Float64(-t_0)
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * abs(J)) * t_2) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_4 = Float64(abs(J) + abs(J))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(t_0 / abs(t_0))) * abs(U));
	elseif (t_3 <= 5e-153)
		tmp = Float64(Float64(t_4 * t_1) * sqrt(Float64(1.0 + Float64(Float64(abs(U) / t_4) * Float64(abs(U) / Float64(Float64(0.5 + 0.5) * t_4))))));
	elseif (t_3 <= 5e+269)
		tmp = Float64(t_4 * Float64(t_1 * sqrt(Float64(Float64(Float64(abs(U) / Float64(Float64(t_4 * t_4) * Float64(0.5 + Float64(0.5 * cos(K))))) * abs(U)) - -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 = -t_0;
	t_2 = cos((K / 2.0));
	t_3 = ((-2.0 * abs(J)) * t_2) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_2)) ^ 2.0)));
	t_4 = abs(J) + abs(J);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (-1.0 * (t_0 / abs(t_0))) * abs(U);
	elseif (t_3 <= 5e-153)
		tmp = (t_4 * t_1) * sqrt((1.0 + ((abs(U) / t_4) * (abs(U) / ((0.5 + 0.5) * t_4)))));
	elseif (t_3 <= 5e+269)
		tmp = t_4 * (t_1 * sqrt((((abs(U) / ((t_4 * t_4) * (0.5 + (0.5 * cos(K))))) * abs(U)) - -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[(-1/2 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[Cos[N[(K / 2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(N[(-1 * N[(t$95$0 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4717453031026927/943490606205385338060388645247067222729230305104110107094051575061406040598037213021531681294414691885367093757690961224942646157481198158140358562858174010912348831744], N[(N[(t$95$4 * t$95$1), $MachinePrecision] * N[Sqrt[N[(1 + N[(N[(N[Abs[U], $MachinePrecision] / t$95$4), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[(1/2 + 1/2), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880], N[(t$95$4 * N[(t$95$1 * N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(1/2 + N[(1/2 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 * N[(1/2 * 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 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 15.1%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 26.7%

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

   9. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.0000000000000003e-153

   1. Initial program 72.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 72.5%

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 72.5%

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.0%

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

   if 5.0000000000000003e-153 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.0000000000000002e269

   1. Initial program 72.6%

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

   3. Applied rewrites 61.3%

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

   if 5.0000000000000002e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{-1}{2} \cdot K\right)\\ t_1 := -t\_0\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \left|J\right| + \left|J\right|\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \frac{t\_0}{\left|t\_0\right|}\right) \cdot \left|U\right|\\ \mathbf{elif}\;t\_3 \leq \frac{4717453031026927}{943490606205385338060388645247067222729230305104110107094051575061406040598037213021531681294414691885367093757690961224942646157481198158140358562858174010912348831744}:\\ \;\;\;\;\left(t\_4 \cdot t\_1\right) \cdot \sqrt{1 + \frac{\left|U\right|}{t\_4} \cdot \frac{\left|U\right|}{\left(\frac{1}{2} + \frac{1}{2}\right) \cdot t\_4}}\\ \mathbf{elif}\;t\_3 \leq 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880:\\ \;\;\;\;\left(\sqrt{\frac{\left|U\right|}{\left(t\_4 \cdot t\_4\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} \cdot \left|U\right| - -1} \cdot t\_4\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -1/2 K)))
       (t_1 (- t_0))
       (t_2 (cos (/ K 2)))
       (t_3
        (*
         (* (* -2 (fabs J)) t_2)
         (sqrt (+ 1 (pow (/ (fabs U) (* (* 2 (fabs J)) t_2)) 2)))))
       (t_4 (+ (fabs J) (fabs J))))
  (*
   (copysign 1 J)
   (if (<= t_3 (- INFINITY))
     (* (* -1 (/ t_0 (fabs t_0))) (fabs U))
     (if (<=
          t_3
          4717453031026927/943490606205385338060388645247067222729230305104110107094051575061406040598037213021531681294414691885367093757690961224942646157481198158140358562858174010912348831744)
       (*
        (* t_4 t_1)
        (sqrt
         (+ 1 (* (/ (fabs U) t_4) (/ (fabs U) (* (+ 1/2 1/2) t_4))))))
       (if (<=
            t_3
            500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880)
         (*
          (*
           (sqrt
            (-
             (*
              (/ (fabs U) (* (* t_4 t_4) (+ 1/2 (* 1/2 (cos K)))))
              (fabs U))
             -1))
           t_4)
          t_1)
         (* 2 (* 1/2 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = -t_0;
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * fabs(J)) * t_2) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_4 = fabs(J) + fabs(J);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (-1.0 * (t_0 / fabs(t_0))) * fabs(U);
	} else if (t_3 <= 5e-153) {
		tmp = (t_4 * t_1) * sqrt((1.0 + ((fabs(U) / t_4) * (fabs(U) / ((0.5 + 0.5) * t_4)))));
	} else if (t_3 <= 5e+269) {
		tmp = (sqrt((((fabs(U) / ((t_4 * t_4) * (0.5 + (0.5 * cos(K))))) * fabs(U)) - -1.0)) * t_4) * t_1;
	} 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 = -t_0;
	double t_2 = Math.cos((K / 2.0));
	double t_3 = ((-2.0 * Math.abs(J)) * t_2) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_2)), 2.0)));
	double t_4 = Math.abs(J) + Math.abs(J);
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (-1.0 * (t_0 / Math.abs(t_0))) * Math.abs(U);
	} else if (t_3 <= 5e-153) {
		tmp = (t_4 * t_1) * Math.sqrt((1.0 + ((Math.abs(U) / t_4) * (Math.abs(U) / ((0.5 + 0.5) * t_4)))));
	} else if (t_3 <= 5e+269) {
		tmp = (Math.sqrt((((Math.abs(U) / ((t_4 * t_4) * (0.5 + (0.5 * Math.cos(K))))) * Math.abs(U)) - -1.0)) * t_4) * t_1;
	} 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 = -t_0
	t_2 = math.cos((K / 2.0))
	t_3 = ((-2.0 * math.fabs(J)) * t_2) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_2)), 2.0)))
	t_4 = math.fabs(J) + math.fabs(J)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (-1.0 * (t_0 / math.fabs(t_0))) * math.fabs(U)
	elif t_3 <= 5e-153:
		tmp = (t_4 * t_1) * math.sqrt((1.0 + ((math.fabs(U) / t_4) * (math.fabs(U) / ((0.5 + 0.5) * t_4)))))
	elif t_3 <= 5e+269:
		tmp = (math.sqrt((((math.fabs(U) / ((t_4 * t_4) * (0.5 + (0.5 * math.cos(K))))) * math.fabs(U)) - -1.0)) * t_4) * t_1
	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 = Float64(-t_0)
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * abs(J)) * t_2) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_4 = Float64(abs(J) + abs(J))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(t_0 / abs(t_0))) * abs(U));
	elseif (t_3 <= 5e-153)
		tmp = Float64(Float64(t_4 * t_1) * sqrt(Float64(1.0 + Float64(Float64(abs(U) / t_4) * Float64(abs(U) / Float64(Float64(0.5 + 0.5) * t_4))))));
	elseif (t_3 <= 5e+269)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(abs(U) / Float64(Float64(t_4 * t_4) * Float64(0.5 + Float64(0.5 * cos(K))))) * abs(U)) - -1.0)) * t_4) * t_1);
	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 = -t_0;
	t_2 = cos((K / 2.0));
	t_3 = ((-2.0 * abs(J)) * t_2) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_2)) ^ 2.0)));
	t_4 = abs(J) + abs(J);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (-1.0 * (t_0 / abs(t_0))) * abs(U);
	elseif (t_3 <= 5e-153)
		tmp = (t_4 * t_1) * sqrt((1.0 + ((abs(U) / t_4) * (abs(U) / ((0.5 + 0.5) * t_4)))));
	elseif (t_3 <= 5e+269)
		tmp = (sqrt((((abs(U) / ((t_4 * t_4) * (0.5 + (0.5 * cos(K))))) * abs(U)) - -1.0)) * t_4) * t_1;
	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[(-1/2 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[Cos[N[(K / 2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(N[(-1 * N[(t$95$0 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4717453031026927/943490606205385338060388645247067222729230305104110107094051575061406040598037213021531681294414691885367093757690961224942646157481198158140358562858174010912348831744], N[(N[(t$95$4 * t$95$1), $MachinePrecision] * N[Sqrt[N[(1 + N[(N[(N[Abs[U], $MachinePrecision] / t$95$4), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[(1/2 + 1/2), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880], N[(N[(N[Sqrt[N[(N[(N[(N[Abs[U], $MachinePrecision] / N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(1/2 + N[(1/2 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$1), $MachinePrecision], N[(2 * N[(1/2 * 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 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 15.1%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 26.7%

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

   9. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.0000000000000003e-153

   1. Initial program 72.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 72.5%

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 72.5%

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.0%

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

   if 5.0000000000000003e-153 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.0000000000000002e269

   1. Initial program 72.6%

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

   3. Applied rewrites 61.3%

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

   if 5.0000000000000002e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{-1}{2} \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \left|J\right| + \left|J\right|\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(-1 \cdot \frac{t\_0}{\left|t\_0\right|}\right) \cdot \left|U\right|\\ \mathbf{elif}\;t\_2 \leq 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880:\\ \;\;\;\;\left(t\_3 \cdot \left(-t\_0\right)\right) \cdot \sqrt{1 + \frac{\left|U\right|}{t\_3} \cdot \frac{\left|U\right|}{\left(\frac{1}{2} + \frac{1}{2}\right) \cdot t\_3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -1/2 K)))
       (t_1 (cos (/ K 2)))
       (t_2
        (*
         (* (* -2 (fabs J)) t_1)
         (sqrt (+ 1 (pow (/ (fabs U) (* (* 2 (fabs J)) t_1)) 2)))))
       (t_3 (+ (fabs J) (fabs J))))
  (*
   (copysign 1 J)
   (if (<= t_2 (- INFINITY))
     (* (* -1 (/ t_0 (fabs t_0))) (fabs U))
     (if (<=
          t_2
          500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880)
       (*
        (* t_3 (- t_0))
        (sqrt
         (+ 1 (* (/ (fabs U) t_3) (/ (fabs U) (* (+ 1/2 1/2) t_3))))))
       (* 2 (* 1/2 (fabs U))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * fabs(J)) * t_1) * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double t_3 = fabs(J) + fabs(J);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (-1.0 * (t_0 / fabs(t_0))) * fabs(U);
	} else if (t_2 <= 5e+269) {
		tmp = (t_3 * -t_0) * sqrt((1.0 + ((fabs(U) / t_3) * (fabs(U) / ((0.5 + 0.5) * t_3)))));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * Math.abs(J)) * t_1) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double t_3 = Math.abs(J) + Math.abs(J);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (-1.0 * (t_0 / Math.abs(t_0))) * Math.abs(U);
	} else if (t_2 <= 5e+269) {
		tmp = (t_3 * -t_0) * Math.sqrt((1.0 + ((Math.abs(U) / t_3) * (Math.abs(U) / ((0.5 + 0.5) * t_3)))));
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * math.fabs(J)) * t_1) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	t_3 = math.fabs(J) + math.fabs(J)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (-1.0 * (t_0 / math.fabs(t_0))) * math.fabs(U)
	elif t_2 <= 5e+269:
		tmp = (t_3 * -t_0) * math.sqrt((1.0 + ((math.fabs(U) / t_3) * (math.fabs(U) / ((0.5 + 0.5) * t_3)))))
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * abs(J)) * t_1) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	t_3 = Float64(abs(J) + abs(J))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 * Float64(t_0 / abs(t_0))) * abs(U));
	elseif (t_2 <= 5e+269)
		tmp = Float64(Float64(t_3 * Float64(-t_0)) * sqrt(Float64(1.0 + Float64(Float64(abs(U) / t_3) * Float64(abs(U) / Float64(Float64(0.5 + 0.5) * t_3))))));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * abs(J)) * t_1) * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	t_3 = abs(J) + abs(J);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (-1.0 * (t_0 / abs(t_0))) * abs(U);
	elseif (t_2 <= 5e+269)
		tmp = (t_3 * -t_0) * sqrt((1.0 + ((abs(U) / t_3) * (abs(U) / ((0.5 + 0.5) * t_3)))));
	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[(-1/2 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(N[(-1 * N[(t$95$0 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[U], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880], N[(N[(t$95$3 * (-t$95$0)), $MachinePrecision] * N[Sqrt[N[(1 + N[(N[(N[Abs[U], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[(1/2 + 1/2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2 * N[(1/2 * 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 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   6. Applied rewrites 15.1%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 26.7%

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

   9. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.0000000000000002e269

   1. Initial program 72.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 72.5%

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 72.5%

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.0%

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

   if 5.0000000000000002e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\frac{-1}{2} \cdot \frac{\left|U\right|}{J}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880:\\ \;\;\;\;\left(\left(J + J\right) \cdot \left(-\cos \left(\frac{-1}{2} \cdot K\right)\right)\right) \cdot \sqrt{1 + \frac{\left|U\right|}{J + J} \cdot \frac{\left|U\right|}{\left(\frac{1}{2} + \frac{1}{2}\right) \cdot \left(J + J\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2)))
       (t_1
        (*
         (* (* -2 J) t_0)
         (sqrt (+ 1 (pow (/ (fabs U) (* (* 2 J) t_0)) 2))))))
  (if (<= t_1 (- INFINITY))
    (* 2 (* J (* -1/2 (/ (fabs U) J))))
    (if (<=
         t_1
         500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880)
      (*
       (* (+ J J) (- (cos (* -1/2 K))))
       (sqrt
        (+
         1
         (*
          (/ (fabs U) (+ J J))
          (/ (fabs U) (* (+ 1/2 1/2) (+ J J)))))))
      (* 2 (* 1/2 (fabs U)))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 2.0 * (J * (-0.5 * (fabs(U) / J)));
	} else if (t_1 <= 5e+269) {
		tmp = ((J + J) * -cos((-0.5 * K))) * sqrt((1.0 + ((fabs(U) / (J + J)) * (fabs(U) / ((0.5 + 0.5) * (J + J))))));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 2.0 * (J * (-0.5 * (Math.abs(U) / J)));
	} else if (t_1 <= 5e+269) {
		tmp = ((J + J) * -Math.cos((-0.5 * K))) * Math.sqrt((1.0 + ((Math.abs(U) / (J + J)) * (Math.abs(U) / ((0.5 + 0.5) * (J + J))))));
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 2.0 * (J * (-0.5 * (math.fabs(U) / J)))
	elif t_1 <= 5e+269:
		tmp = ((J + J) * -math.cos((-0.5 * K))) * math.sqrt((1.0 + ((math.fabs(U) / (J + J)) * (math.fabs(U) / ((0.5 + 0.5) * (J + J))))))
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(2.0 * Float64(J * Float64(-0.5 * Float64(abs(U) / J))));
	elseif (t_1 <= 5e+269)
		tmp = Float64(Float64(Float64(J + J) * Float64(-cos(Float64(-0.5 * K)))) * sqrt(Float64(1.0 + Float64(Float64(abs(U) / Float64(J + J)) * Float64(abs(U) / Float64(Float64(0.5 + 0.5) * Float64(J + J)))))));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 2.0 * (J * (-0.5 * (abs(U) / J)));
	elseif (t_1 <= 5e+269)
		tmp = ((J + J) * -cos((-0.5 * K))) * sqrt((1.0 + ((abs(U) / (J + J)) * (abs(U) / ((0.5 + 0.5) * (J + J))))));
	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), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(2 * N[(J * N[(-1/2 * N[(N[Abs[U], $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880], N[(N[(N[(J + J), $MachinePrecision] * (-N[Cos[N[(-1/2 * K), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[Sqrt[N[(1 + N[(N[(N[Abs[U], $MachinePrecision] / N[(J + J), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[(N[(1/2 + 1/2), $MachinePrecision] * N[(J + J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2 * N[(1/2 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 13.3%

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 20.5%

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

   10. Applied rewrites 20.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.0000000000000002e269

   1. Initial program 72.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   3. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 72.5%

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

      11. lift-cos.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 72.5%

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.0%

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

   if 5.0000000000000002e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\frac{-1}{2} \cdot \frac{\left|U\right|}{J}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (/ K 2)))
       (t_1
        (*
         (* (* -2 J) t_0)
         (sqrt (+ 1 (pow (/ (fabs U) (* (* 2 J) t_0)) 2))))))
  (if (<= t_1 (- INFINITY))
    (* 2 (* J (* -1/2 (/ (fabs U) J))))
    (if (<=
         t_1
         500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880)
      (* -2 (* J (cos (* 1/2 K))))
      (* 2 (* 1/2 (fabs U)))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((fabs(U) / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 2.0 * (J * (-0.5 * (fabs(U) / J)));
	} else if (t_1 <= 5e+269) {
		tmp = -2.0 * (J * cos((0.5 * K)));
	} else {
		tmp = 2.0 * (0.5 * fabs(U));
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 2.0 * (J * (-0.5 * (Math.abs(U) / J)));
	} else if (t_1 <= 5e+269) {
		tmp = -2.0 * (J * Math.cos((0.5 * K)));
	} else {
		tmp = 2.0 * (0.5 * Math.abs(U));
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 2.0 * (J * (-0.5 * (math.fabs(U) / J)))
	elif t_1 <= 5e+269:
		tmp = -2.0 * (J * math.cos((0.5 * K)))
	else:
		tmp = 2.0 * (0.5 * math.fabs(U))
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(2.0 * Float64(J * Float64(-0.5 * Float64(abs(U) / J))));
	elseif (t_1 <= 5e+269)
		tmp = Float64(-2.0 * Float64(J * cos(Float64(0.5 * K))));
	else
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((abs(U) / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 2.0 * (J * (-0.5 * (abs(U) / J)));
	elseif (t_1 <= 5e+269)
		tmp = -2.0 * (J * cos((0.5 * K)));
	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), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(2 * N[(J * N[(-1/2 * N[(N[Abs[U], $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 500000000000000023376909442728063994594802715665205143420682436372008219697277947305184129090151668469538444067022475144663084092331215165737156638708489908193694639932318967793498760119176155511330039146864335692596466630531151717376319013390688774370983942319641722880], N[(-2 * N[(J * N[Cos[N[(1/2 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 * N[(1/2 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 13.3%

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 20.5%

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

   10. Applied rewrites 20.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.0000000000000002e269

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

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

   11. Taylor expanded in J around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-*.f64 51.5%

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

   13. Applied rewrites 51.5%

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

   if 5.0000000000000002e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(\left|J\right| \cdot \left(\frac{-1}{2} \cdot \frac{\left|U\right|}{\left|J\right|}\right)\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq \frac{-7307508186654515}{365375409332725729550921208179070754913983135744}:\\ \;\;\;\;\left(\left(\left(\left(\left(\frac{1}{384} \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot K\right) \cdot K - -1\right) \cdot \left|J\right|\right) \cdot -2\right) \cdot \sqrt{1}\\ \mathbf{elif}\;t\_2 \leq \frac{-1612226962694291}{4030567406735727282351225165683873035574701889313671280883489296162978382543372035785043761349923613698382530160958318167742519832631573007587730243400112738864034149162331269597866193210040596412843573823632724030670381872189039145190406026970187961498554846937088}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (* 2 (* (fabs J) (* -1/2 (/ (fabs U) (fabs J))))))
       (t_1 (cos (/ K 2)))
       (t_2
        (*
         (* (* -2 (fabs J)) t_1)
         (sqrt (+ 1 (pow (/ (fabs U) (* (* 2 (fabs J)) t_1)) 2))))))
  (*
   (copysign 1 J)
   (if (<= t_2 (- INFINITY))
     t_0
     (if (<=
          t_2
          -7307508186654515/365375409332725729550921208179070754913983135744)
       (*
        (*
         (* (- (* (* (- (* 1/384 (* K K)) 1/8) K) K) -1) (fabs J))
         -2)
        (sqrt 1))
       (if (<=
            t_2
            -1612226962694291/4030567406735727282351225165683873035574701889313671280883489296162978382543372035785043761349923613698382530160958318167742519832631573007587730243400112738864034149162331269597866193210040596412843573823632724030670381872189039145190406026970187961498554846937088)
         t_0
         (* 2 (* 1/2 (fabs U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = 2.0 * (fabs(J) * (-0.5 * (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 = t_0;
	} else if (t_2 <= -2e-32) {
		tmp = (((((((0.0026041666666666665 * (K * K)) - 0.125) * K) * K) - -1.0) * fabs(J)) * -2.0) * sqrt(1.0);
	} else if (t_2 <= -4e-250) {
		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 = 2.0 * (Math.abs(J) * (-0.5 * (Math.abs(U) / Math.abs(J))));
	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 = t_0;
	} else if (t_2 <= -2e-32) {
		tmp = (((((((0.0026041666666666665 * (K * K)) - 0.125) * K) * K) - -1.0) * Math.abs(J)) * -2.0) * Math.sqrt(1.0);
	} else if (t_2 <= -4e-250) {
		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 = 2.0 * (math.fabs(J) * (-0.5 * (math.fabs(U) / math.fabs(J))))
	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 = t_0
	elif t_2 <= -2e-32:
		tmp = (((((((0.0026041666666666665 * (K * K)) - 0.125) * K) * K) - -1.0) * math.fabs(J)) * -2.0) * math.sqrt(1.0)
	elif t_2 <= -4e-250:
		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(2.0 * Float64(abs(J) * Float64(-0.5 * 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 = t_0;
	elseif (t_2 <= -2e-32)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.0026041666666666665 * Float64(K * K)) - 0.125) * K) * K) - -1.0) * abs(J)) * -2.0) * sqrt(1.0));
	elseif (t_2 <= -4e-250)
		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 = 2.0 * (abs(J) * (-0.5 * (abs(U) / abs(J))));
	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 = t_0;
	elseif (t_2 <= -2e-32)
		tmp = (((((((0.0026041666666666665 * (K * K)) - 0.125) * K) * K) - -1.0) * abs(J)) * -2.0) * sqrt(1.0);
	elseif (t_2 <= -4e-250)
		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[(2 * N[(N[Abs[J], $MachinePrecision] * N[(-1/2 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(2 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, -7307508186654515/365375409332725729550921208179070754913983135744], N[(N[(N[(N[(N[(N[(N[(N[(1/384 * N[(K * K), $MachinePrecision]), $MachinePrecision] - 1/8), $MachinePrecision] * K), $MachinePrecision] * K), $MachinePrecision] - -1), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -2), $MachinePrecision] * N[Sqrt[1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1612226962694291/4030567406735727282351225165683873035574701889313671280883489296162978382543372035785043761349923613698382530160958318167742519832631573007587730243400112738864034149162331269597866193210040596412843573823632724030670381872189039145190406026970187961498554846937088], t$95$0, N[(2 * N[(1/2 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or -2.0000000000000001e-32 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000002e-250

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 13.3%

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 20.5%

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

   10. Applied rewrites 20.5%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e-32

   1. Initial program 72.6%

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

   2. Taylor expanded in J around inf

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{1} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + \color{blue}{{K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \color{blue}{\left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)}\right)\right) \cdot \sqrt{1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\color{blue}{\frac{1}{384} \cdot {K}^{2}} - \frac{1}{8}\right)\right)\right) \cdot \sqrt{1} \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \color{blue}{\frac{1}{8}}\right)\right)\right) \cdot \sqrt{1} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot \sqrt{1} \]

      6. lower-pow.f64 26.8%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot \sqrt{1} \]

   7. Applied rewrites 26.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{1} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right)} \cdot \sqrt{1} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(-2 \cdot J\right)} \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot \sqrt{1} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left(J \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right)\right)} \cdot \sqrt{1} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(J \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot -2\right)} \cdot \sqrt{1} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(J \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right) \cdot -2\right)} \cdot \sqrt{1} \]

   9. Applied rewrites 26.8%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\frac{1}{384} \cdot \left(K \cdot K\right) - \frac{1}{8}\right) \cdot K\right) \cdot K - -1\right) \cdot J\right) \cdot -2\right)} \cdot \sqrt{1} \]

   if -4.0000000000000002e-250 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in K around 0

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

      1. lower-*.f64 27.3%

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

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 43.3% accurate, 1.5× speedup

Math

\[\mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq \frac{-2656869547074365}{241533595188578646543459493183249209018682958106652187416077203215719946393097526533512110411370161122653976001968886073585317416315186728483931792091692546793561300926464}:\\ \;\;\;\;2 \cdot \left(\frac{1}{2} \cdot \left|U\right|\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left|J\right| \cdot \left(\frac{-1}{2} \cdot \frac{\left|U\right|}{\left|J\right|}\right)\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (*
 (copysign 1 J)
 (if (<=
      (cos (/ K 2))
      -2656869547074365/241533595188578646543459493183249209018682958106652187416077203215719946393097526533512110411370161122653976001968886073585317416315186728483931792091692546793561300926464)
   (* 2 (* 1/2 (fabs U)))
   (* 2 (* (fabs J) (* -1/2 (/ (fabs U) (fabs J))))))))

C

double code(double J, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -1.1e-155) {
		tmp = 2.0 * (0.5 * fabs(U));
	} else {
		tmp = 2.0 * (fabs(J) * (-0.5 * (fabs(U) / fabs(J))));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -1.1e-155) {
		tmp = 2.0 * (0.5 * Math.abs(U));
	} else {
		tmp = 2.0 * (Math.abs(J) * (-0.5 * (Math.abs(U) / Math.abs(J))));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -1.1e-155:
		tmp = 2.0 * (0.5 * math.fabs(U))
	else:
		tmp = 2.0 * (math.fabs(J) * (-0.5 * (math.fabs(U) / math.fabs(J))))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -1.1e-155)
		tmp = Float64(2.0 * Float64(0.5 * abs(U)));
	else
		tmp = Float64(2.0 * Float64(abs(J) * Float64(-0.5 * Float64(abs(U) / abs(J)))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -1.1e-155)
		tmp = 2.0 * (0.5 * abs(U));
	else
		tmp = 2.0 * (abs(J) * (-0.5 * (abs(U) / abs(J))));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Cos[N[(K / 2), $MachinePrecision]], $MachinePrecision], -2656869547074365/241533595188578646543459493183249209018682958106652187416077203215719946393097526533512110411370161122653976001968886073585317416315186728483931792091692546793561300926464], N[(2 * N[(1/2 * N[Abs[U], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2 * N[(N[Abs[J], $MachinePrecision] * N[(-1/2 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -1.1e-155

   1. Initial program 72.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 13.4%

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

   5. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

         \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

      9. lower-*.f64 27.0%

         \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

   7. Applied rewrites 27.0%

      \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 27.3%

         \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]

   10. Applied rewrites 27.3%

       \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]

   if -1.1e-155 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 72.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right)\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{4}}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

   4. Applied rewrites 13.4%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \color{blue}{\sqrt{\frac{\frac{1}{4}}{{J}^{2}}}}\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2}}}\right)\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2}}}\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2}}}\right)\right) \]

      4. lower-pow.f64 13.3%

         \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2}}}\right)\right) \]

   7. Applied rewrites 13.3%

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \color{blue}{\sqrt{\frac{\frac{1}{4}}{{J}^{2}}}}\right)\right) \]

   8. Taylor expanded in J around -inf

      \[\leadsto 2 \cdot \left(J \cdot \left(\frac{-1}{2} \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(J \cdot \left(\frac{-1}{2} \cdot \frac{U}{J}\right)\right) \]

      2. lower-/.f64 20.5%

         \[\leadsto 2 \cdot \left(J \cdot \left(\frac{-1}{2} \cdot \frac{U}{J}\right)\right) \]

   10. Applied rewrites 20.5%

       \[\leadsto 2 \cdot \left(J \cdot \left(\frac{-1}{2} \cdot \frac{U}{\color{blue}{J}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 27.3% accurate, 33.9× speedup

Math

\[2 \cdot \left(\frac{1}{2} \cdot U\right) \]

FPCore

(FPCore (J K U)
  :precision binary64
  (* 2 (* 1/2 U)))

C

double code(double J, double K, double U) {
	return 2.0 * (0.5 * U);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = 2.0d0 * (0.5d0 * u)
end function

Java

public static double code(double J, double K, double U) {
	return 2.0 * (0.5 * U);
}

Python

def code(J, K, U):
	return 2.0 * (0.5 * U)

Julia

function code(J, K, U)
	return Float64(2.0 * Float64(0.5 * U))
end

MATLAB

function tmp = code(J, K, U)
	tmp = 2.0 * (0.5 * U);
end

Wolfram

code[J_, K_, U_] := N[(2 * N[(1/2 * U), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.6%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

2. Taylor expanded in U around -inf

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right)\right)\right) \]

   5. lower-cos.f64 N/A

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right)\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{4}}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

   8. lower-/.f64 N/A

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

   10. lower-pow.f64 N/A

       \[\leadsto 2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right) \]

4. Applied rewrites 13.4%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right)\right)} \]

5. Taylor expanded in J around 0

   \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

   3. lower-cos.f64 N/A

      \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

   6. lower-/.f64 N/A

      \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

   8. lower-cos.f64 N/A

      \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

   9. lower-*.f64 27.0%

      \[\leadsto 2 \cdot \left(U \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)\right) \]

7. Applied rewrites 27.0%

   \[\leadsto 2 \cdot \left(U \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}\right) \]

8. Taylor expanded in K around 0

   \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
9. Step-by-step derivation

   1. lower-*.f64 27.3%

      \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]

10. Applied rewrites 27.3%

    \[\leadsto 2 \cdot \left(\frac{1}{2} \cdot U\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))