Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.2% → 99.7%

Time: 7.4s

Alternatives: 14

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (cos (* 0.5 K)))
        (t_3 (* -2.0 (* U_m (* t_2 (sqrt (/ 0.25 (pow t_2 2.0))))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      t_3
      (if (<= t_1 2e+306)
        (*
         (* (* (cos (* -0.5 K)) J_m) -2.0)
         (sqrt (- (pow (/ U_m (* (+ J_m J_m) t_2)) 2.0) -1.0)))
        t_3)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = cos((0.5 * K));
	double t_3 = -2.0 * (U_m * (t_2 * sqrt((0.25 / pow(t_2, 2.0)))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 2e+306) {
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * sqrt((pow((U_m / ((J_m + J_m) * t_2)), 2.0) - -1.0));
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = Math.cos((0.5 * K));
	double t_3 = -2.0 * (U_m * (t_2 * Math.sqrt((0.25 / Math.pow(t_2, 2.0)))));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_1 <= 2e+306) {
		tmp = ((Math.cos((-0.5 * K)) * J_m) * -2.0) * Math.sqrt((Math.pow((U_m / ((J_m + J_m) * t_2)), 2.0) - -1.0));
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	t_2 = math.cos((0.5 * K))
	t_3 = -2.0 * (U_m * (t_2 * math.sqrt((0.25 / math.pow(t_2, 2.0)))))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_3
	elif t_1 <= 2e+306:
		tmp = ((math.cos((-0.5 * K)) * J_m) * -2.0) * math.sqrt((math.pow((U_m / ((J_m + J_m) * t_2)), 2.0) - -1.0))
	else:
		tmp = t_3
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	t_3 = Float64(-2.0 * Float64(U_m * Float64(t_2 * sqrt(Float64(0.25 / (t_2 ^ 2.0))))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+306)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J_m) * -2.0) * sqrt(Float64((Float64(U_m / Float64(Float64(J_m + J_m) * t_2)) ^ 2.0) - -1.0)));
	else
		tmp = t_3;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	t_2 = cos((0.5 * K));
	t_3 = -2.0 * (U_m * (t_2 * sqrt((0.25 / (t_2 ^ 2.0)))));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_3;
	elseif (t_1 <= 2e+306)
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * sqrt((((U_m / ((J_m + J_m) * t_2)) ^ 2.0) - -1.0));
	else
		tmp = t_3;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[(U$95$m * N[(t$95$2 * N[Sqrt[N[(0.25 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+306], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(J$95$m + J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   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 24.6%

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

   5. Taylor expanded in J around 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 52.3

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

   7. Applied rewrites 52.3%

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

   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))))) < 2.00000000000000003e306

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

   3. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 73.2%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (cos (* 0.5 K)))
        (t_3 (* -2.0 (* U_m (* t_2 (sqrt (/ 0.25 (pow t_2 2.0))))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      t_3
      (if (<= t_1 2e+306)
        (*
         (*
          (sqrt
           (-
            (/ (* (/ (/ U_m (fma (cos K) 0.5 0.5)) J_m) (/ U_m J_m)) 4.0)
            -1.0))
          (cos (* -0.5 K)))
         (* J_m -2.0))
        t_3)))))

C

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

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	t_3 = Float64(-2.0 * Float64(U_m * Float64(t_2 * sqrt(Float64(0.25 / (t_2 ^ 2.0))))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+306)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(U_m / fma(cos(K), 0.5, 0.5)) / J_m) * Float64(U_m / J_m)) / 4.0) - -1.0)) * cos(Float64(-0.5 * K))) * Float64(J_m * -2.0));
	else
		tmp = t_3;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[(U$95$m * N[(t$95$2 * N[Sqrt[N[(0.25 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+306], N[(N[(N[Sqrt[N[(N[(N[(N[(N[(U$95$m / N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], t$95$3]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   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 24.6%

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

   5. Taylor expanded in J around 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 52.3

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

   7. Applied rewrites 52.3%

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

   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))))) < 2.00000000000000003e306

   1. Initial program 73.2%

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 73.0%

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

Alternative 3: 99.5% accurate, 0.3× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (cos (* 0.5 K)))
        (t_3 (* -2.0 (* U_m (* t_2 (sqrt (/ 0.25 (pow t_2 2.0))))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      t_3
      (if (<= t_1 2e+306)
        (*
         (*
          (sqrt
           (fma
            (/ U_m J_m)
            (* (/ U_m (* (fma (cos K) 0.5 0.5) J_m)) 0.25)
            1.0))
          (* J_m -2.0))
         (cos (* -0.5 K)))
        t_3)))))

C

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

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	t_3 = Float64(-2.0 * Float64(U_m * Float64(t_2 * sqrt(Float64(0.25 / (t_2 ^ 2.0))))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+306)
		tmp = Float64(Float64(sqrt(fma(Float64(U_m / J_m), Float64(Float64(U_m / Float64(fma(cos(K), 0.5, 0.5) * J_m)) * 0.25), 1.0)) * Float64(J_m * -2.0)) * cos(Float64(-0.5 * K)));
	else
		tmp = t_3;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[(U$95$m * N[(t$95$2 * N[Sqrt[N[(0.25 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+306], N[(N[(N[Sqrt[N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(N[(U$95$m / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   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 24.6%

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

   5. Taylor expanded in J around 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 52.3

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

   7. Applied rewrites 52.3%

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

   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))))) < 2.00000000000000003e306

   1. Initial program 73.2%

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 73.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 73.1

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 73.1

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

   7. Applied rewrites 73.1%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, \frac{U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J} \cdot 0.25, 1\right)}} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(-0.5 \cdot K\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(0.5 \cdot K\right)\\ t_3 := -2 \cdot \left(U\_m \cdot \left(t\_2 \cdot \sqrt{\frac{0.25}{{t\_2}^{2}}}\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+307}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J\_m} \cdot U\_m, \frac{0.25}{J\_m}, 1\right)} \cdot \left(J\_m \cdot -2\right)\right) \cdot \cos \left(-0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (cos (* 0.5 K)))
        (t_3 (* -2.0 (* U_m (* t_2 (sqrt (/ 0.25 (pow t_2 2.0))))))))
   (*
    J_s
    (if (<= t_1 -2e+307)
      t_3
      (if (<= t_1 2e+269)
        (*
         (*
          (sqrt
           (fma
            (* (/ U_m (* (fma (cos K) 0.5 0.5) J_m)) U_m)
            (/ 0.25 J_m)
            1.0))
          (* J_m -2.0))
         (cos (* -0.5 K)))
        t_3)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = cos((0.5 * K));
	double t_3 = -2.0 * (U_m * (t_2 * sqrt((0.25 / pow(t_2, 2.0)))));
	double tmp;
	if (t_1 <= -2e+307) {
		tmp = t_3;
	} else if (t_1 <= 2e+269) {
		tmp = (sqrt(fma(((U_m / (fma(cos(K), 0.5, 0.5) * J_m)) * U_m), (0.25 / J_m), 1.0)) * (J_m * -2.0)) * cos((-0.5 * K));
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	t_3 = Float64(-2.0 * Float64(U_m * Float64(t_2 * sqrt(Float64(0.25 / (t_2 ^ 2.0))))))
	tmp = 0.0
	if (t_1 <= -2e+307)
		tmp = t_3;
	elseif (t_1 <= 2e+269)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / Float64(fma(cos(K), 0.5, 0.5) * J_m)) * U_m), Float64(0.25 / J_m), 1.0)) * Float64(J_m * -2.0)) * cos(Float64(-0.5 * K)));
	else
		tmp = t_3;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[(U$95$m * N[(t$95$2 * N[Sqrt[N[(0.25 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+307], t$95$3, If[LessEqual[t$95$1, 2e+269], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision] * N[(0.25 / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999997e307 or 2.0000000000000001e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   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 24.6%

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

   5. Taylor expanded in J around 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 52.3

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

   7. Applied rewrites 52.3%

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

   if -1.99999999999999997e307 < (*.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))))) < 2.0000000000000001e269

   1. Initial program 73.2%

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 73.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. associate-*l/ N/A

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

      11. times-frac N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   7. Applied rewrites 70.1%

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J} \cdot U, \frac{0.25}{J}, 1\right)}} \cdot \left(J \cdot -2\right)\right) \cdot \cos \left(-0.5 \cdot K\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(0.5 \cdot K\right)\\ t_3 := -2 \cdot \left(U\_m \cdot \left(t\_2 \cdot \sqrt{\frac{0.25}{{t\_2}^{2}}}\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}}{4} - -1} \cdot \left(J\_m \cdot -2\right)\right) \cdot \cos \left(-0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (cos (* 0.5 K)))
        (t_3 (* -2.0 (* U_m (* t_2 (sqrt (/ 0.25 (pow t_2 2.0))))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      t_3
      (if (<= t_1 2e+269)
        (*
         (* (sqrt (- (/ (* (/ U_m J_m) (/ U_m J_m)) 4.0) -1.0)) (* J_m -2.0))
         (cos (* -0.5 K)))
        t_3)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = cos((0.5 * K));
	double t_3 = -2.0 * (U_m * (t_2 * sqrt((0.25 / pow(t_2, 2.0)))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 2e+269) {
		tmp = (sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)) * (J_m * -2.0)) * cos((-0.5 * K));
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = Math.cos((0.5 * K));
	double t_3 = -2.0 * (U_m * (t_2 * Math.sqrt((0.25 / Math.pow(t_2, 2.0)))));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_1 <= 2e+269) {
		tmp = (Math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)) * (J_m * -2.0)) * Math.cos((-0.5 * K));
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	t_2 = math.cos((0.5 * K))
	t_3 = -2.0 * (U_m * (t_2 * math.sqrt((0.25 / math.pow(t_2, 2.0)))))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_3
	elif t_1 <= 2e+269:
		tmp = (math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)) * (J_m * -2.0)) * math.cos((-0.5 * K))
	else:
		tmp = t_3
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	t_3 = Float64(-2.0 * Float64(U_m * Float64(t_2 * sqrt(Float64(0.25 / (t_2 ^ 2.0))))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+269)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) / 4.0) - -1.0)) * Float64(J_m * -2.0)) * cos(Float64(-0.5 * K)));
	else
		tmp = t_3;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	t_2 = cos((0.5 * K));
	t_3 = -2.0 * (U_m * (t_2 * sqrt((0.25 / (t_2 ^ 2.0)))));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_3;
	elseif (t_1 <= 2e+269)
		tmp = (sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)) * (J_m * -2.0)) * cos((-0.5 * K));
	else
		tmp = t_3;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[(U$95$m * N[(t$95$2 * N[Sqrt[N[(0.25 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+269], N[(N[(N[Sqrt[N[(N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000001e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   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 24.6%

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

   5. Taylor expanded in J around 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 52.3

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

   7. Applied rewrites 52.3%

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

   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))))) < 2.0000000000000001e269

   1. Initial program 73.2%

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.1%

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

Alternative 6: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ t_3 := -2 \cdot \left(U\_m \cdot \left(t\_2 \cdot \sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(-1 \cdot K\right)}}\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}}{4} - -1} \cdot \left(J\_m \cdot -2\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (cos (* -0.5 K)))
        (t_3
         (*
          -2.0
          (* U_m (* t_2 (sqrt (/ 0.25 (+ 0.5 (* 0.5 (cos (* -1.0 K)))))))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      t_3
      (if (<= t_1 2e+269)
        (*
         (* (sqrt (- (/ (* (/ U_m J_m) (/ U_m J_m)) 4.0) -1.0)) (* J_m -2.0))
         t_2)
        t_3)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = cos((-0.5 * K));
	double t_3 = -2.0 * (U_m * (t_2 * sqrt((0.25 / (0.5 + (0.5 * cos((-1.0 * K))))))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 2e+269) {
		tmp = (sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)) * (J_m * -2.0)) * t_2;
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = Math.cos((-0.5 * K));
	double t_3 = -2.0 * (U_m * (t_2 * Math.sqrt((0.25 / (0.5 + (0.5 * Math.cos((-1.0 * K))))))));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_1 <= 2e+269) {
		tmp = (Math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)) * (J_m * -2.0)) * t_2;
	} else {
		tmp = t_3;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	t_2 = math.cos((-0.5 * K))
	t_3 = -2.0 * (U_m * (t_2 * math.sqrt((0.25 / (0.5 + (0.5 * math.cos((-1.0 * K))))))))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t_3
	elif t_1 <= 2e+269:
		tmp = (math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)) * (J_m * -2.0)) * t_2
	else:
		tmp = t_3
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(-0.5 * K))
	t_3 = Float64(-2.0 * Float64(U_m * Float64(t_2 * sqrt(Float64(0.25 / Float64(0.5 + Float64(0.5 * cos(Float64(-1.0 * K)))))))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+269)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) / 4.0) - -1.0)) * Float64(J_m * -2.0)) * t_2);
	else
		tmp = t_3;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	t_2 = cos((-0.5 * K));
	t_3 = -2.0 * (U_m * (t_2 * sqrt((0.25 / (0.5 + (0.5 * cos((-1.0 * K))))))));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t_3;
	elseif (t_1 <= 2e+269)
		tmp = (sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)) * (J_m * -2.0)) * t_2;
	else
		tmp = t_3;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[(U$95$m * N[(t$95$2 * N[Sqrt[N[(0.25 / N[(0.5 + N[(0.5 * N[Cos[N[(-1.0 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+269], N[(N[(N[Sqrt[N[(N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], t$95$3]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000001e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   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 24.6%

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

   6. Applied rewrites 24.3%

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

   7. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 52.1

          \[\leadsto -2 \cdot \left(U \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos \left(-1 \cdot K\right)}}\right)\right) \]

   9. Applied rewrites 52.1%

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

   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))))) < 2.0000000000000001e269

   1. Initial program 73.2%

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.1%

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

Alternative 7: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{-0.5}{J\_m} \cdot U\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, 0.25 \cdot \left(J\_m \cdot {K}^{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{-0.5}{\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\_m} \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
         2e+306)
      (* (* (* (cos (* -0.5 K)) J_m) -2.0) (cosh (asinh (* (/ -0.5 J_m) U_m))))
      (*
       (fma -2.0 J_m (* 0.25 (* J_m (pow K 2.0))))
       (cosh
        (asinh (* (/ -0.5 (* (+ 1.0 (* -0.125 (pow K 2.0))) J_m)) U_m))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= 2e+306) {
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * cosh(asinh(((-0.5 / J_m) * U_m)));
	} else {
		tmp = fma(-2.0, J_m, (0.25 * (J_m * pow(K, 2.0)))) * cosh(asinh(((-0.5 / ((1.0 + (-0.125 * pow(K, 2.0))) * J_m)) * U_m)));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0)))) <= 2e+306)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J_m) * -2.0) * cosh(asinh(Float64(Float64(-0.5 / J_m) * U_m))));
	else
		tmp = Float64(fma(-2.0, J_m, Float64(0.25 * Float64(J_m * (K ^ 2.0)))) * cosh(asinh(Float64(Float64(-0.5 / Float64(Float64(1.0 + Float64(-0.125 * (K ^ 2.0))) * J_m)) * U_m))));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(N[(-0.5 / J$95$m), $MachinePrecision] * U$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J$95$m + N[(0.25 * N[(J$95$m * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(N[(-0.5 / N[(N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.00000000000000003e306

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

   3. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-/.f64 71.0

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

   8. Applied rewrites 71.0%

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

   if 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 43.1

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

   6. Applied rewrites 43.1%

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

   7. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{\frac{-1}{2}}{\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot J} \cdot U\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{\frac{-1}{2}}{\left(1 + \color{blue}{\frac{-1}{8} \cdot {K}^{2}}\right) \cdot J} \cdot U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{\frac{-1}{2}}{\left(1 + \frac{-1}{8} \cdot \color{blue}{{K}^{2}}\right) \cdot J} \cdot U\right) \]

      3. lower-pow.f64 45.8

         \[\leadsto \mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{-0.5}{\left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) \cdot J} \cdot U\right) \]

   9. Applied rewrites 45.8%

      \[\leadsto \mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{-0.5}{\color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} \cdot J} \cdot U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{-0.5}{J\_m} \cdot U\_m\right)\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (* (* (* (cos (* -0.5 K)) J_m) -2.0) (cosh (asinh (* (/ -0.5 J_m) U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (((cos((-0.5 * K)) * J_m) * -2.0) * cosh(asinh(((-0.5 / J_m) * U_m))));
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * (((math.cos((-0.5 * K)) * J_m) * -2.0) * math.cosh(math.asinh(((-0.5 / J_m) * U_m))))

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J_m) * -2.0) * cosh(asinh(Float64(Float64(-0.5 / J_m) * U_m)))))
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * (((cos((-0.5 * K)) * J_m) * -2.0) * cosh(asinh(((-0.5 / J_m) * U_m))));
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(N[(-0.5 / J$95$m), $MachinePrecision] * U$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-neg-rev N/A

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

   7. cosh-asinh-rev N/A

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

   8. lower-cosh.f64 N/A

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

   9. lower-asinh.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. *-commutative N/A

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

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

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

   14. distribute-frac-neg2 N/A

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

3. Applied rewrites 84.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-cos.f64 N/A

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

   8. cos-neg-rev N/A

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

   9. lower-cos.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 84.6

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

5. Applied rewrites 84.6%

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

6. Taylor expanded in K around 0

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

   1. lower-/.f64 71.0

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

8. Applied rewrites 71.0%

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

Alternative 9: 70.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, 0.25 \cdot \left(J\_m \cdot {K}^{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{-0.5}{J\_m} \cdot U\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}}{4} - -1} \cdot \left(J\_m \cdot -2\right)\right) \cdot \cos \left(-0.5 \cdot K\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
         (- INFINITY))
      (*
       (fma -2.0 J_m (* 0.25 (* J_m (pow K 2.0))))
       (cosh (asinh (* (/ -0.5 J_m) U_m))))
      (*
       (* (sqrt (- (/ (* (/ U_m J_m) (/ U_m J_m)) 4.0) -1.0)) (* J_m -2.0))
       (cos (* -0.5 K)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -((double) INFINITY)) {
		tmp = fma(-2.0, J_m, (0.25 * (J_m * pow(K, 2.0)))) * cosh(asinh(((-0.5 / J_m) * U_m)));
	} else {
		tmp = (sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)) * (J_m * -2.0)) * cos((-0.5 * K));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(fma(-2.0, J_m, Float64(0.25 * Float64(J_m * (K ^ 2.0)))) * cosh(asinh(Float64(Float64(-0.5 / J_m) * U_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) / 4.0) - -1.0)) * Float64(J_m * -2.0)) * cos(Float64(-0.5 * K)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(-2.0 * J$95$m + N[(0.25 * N[(J$95$m * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(N[(-0.5 / J$95$m), $MachinePrecision] * U$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 43.1

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

   6. Applied rewrites 43.1%

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

   7. Taylor expanded in K around 0

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

      1. lower-/.f64 43.0

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

   9. Applied rewrites 43.0%

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

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

   1. Initial program 73.2%

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

   3. Applied rewrites 73.1%

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.1%

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

Alternative 10: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 770:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, 0.25 \cdot \left(J\_m \cdot {K}^{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{-0.5}{J\_m} \cdot U\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= K 770.0)
    (*
     (fma -2.0 J_m (* 0.25 (* J_m (pow K 2.0))))
     (cosh (asinh (* (/ -0.5 J_m) U_m))))
    (* (cos (* 0.5 K)) (* -2.0 J_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (K <= 770.0) {
		tmp = fma(-2.0, J_m, (0.25 * (J_m * pow(K, 2.0)))) * cosh(asinh(((-0.5 / J_m) * U_m)));
	} else {
		tmp = cos((0.5 * K)) * (-2.0 * J_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (K <= 770.0)
		tmp = Float64(fma(-2.0, J_m, Float64(0.25 * Float64(J_m * (K ^ 2.0)))) * cosh(asinh(Float64(Float64(-0.5 / J_m) * U_m))));
	else
		tmp = Float64(cos(Float64(0.5 * K)) * Float64(-2.0 * J_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[K, 770.0], N[(N[(-2.0 * J$95$m + N[(0.25 * N[(J$95$m * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(N[(-0.5 / J$95$m), $MachinePrecision] * U$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if K < 770

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 43.1

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

   6. Applied rewrites 43.1%

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

   7. Taylor expanded in K around 0

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

      1. lower-/.f64 43.0

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

   9. Applied rewrites 43.0%

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

   if 770 < K

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

   3. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in J around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

Alternative 11: 58.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(-2 \cdot J\_m\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \left(U\_m \cdot \sqrt{\frac{0.25}{{J\_m}^{2}}}\right)\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-145}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \sqrt{1 + 0.25 \cdot \frac{{U\_m}^{2}}{{J\_m}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (cos (* 0.5 K)) (* -2.0 J_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* -2.0 (* J_m (* U_m (sqrt (/ 0.25 (pow J_m 2.0))))))
      (if (<= t_2 -2e+167)
        t_0
        (if (<= t_2 -1e-145)
          (*
           -2.0
           (* J_m (sqrt (+ 1.0 (* 0.25 (/ (pow U_m 2.0) (pow J_m 2.0)))))))
          t_0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((0.5 * K)) * (-2.0 * J_m);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (J_m * (U_m * sqrt((0.25 / pow(J_m, 2.0)))));
	} else if (t_2 <= -2e+167) {
		tmp = t_0;
	} else if (t_2 <= -1e-145) {
		tmp = -2.0 * (J_m * sqrt((1.0 + (0.25 * (pow(U_m, 2.0) / pow(J_m, 2.0))))));
	} else {
		tmp = t_0;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((0.5 * K)) * (-2.0 * J_m);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (J_m * (U_m * Math.sqrt((0.25 / Math.pow(J_m, 2.0)))));
	} else if (t_2 <= -2e+167) {
		tmp = t_0;
	} else if (t_2 <= -1e-145) {
		tmp = -2.0 * (J_m * Math.sqrt((1.0 + (0.25 * (Math.pow(U_m, 2.0) / Math.pow(J_m, 2.0))))));
	} else {
		tmp = t_0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((0.5 * K)) * (-2.0 * J_m)
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (J_m * (U_m * math.sqrt((0.25 / math.pow(J_m, 2.0)))))
	elif t_2 <= -2e+167:
		tmp = t_0
	elif t_2 <= -1e-145:
		tmp = -2.0 * (J_m * math.sqrt((1.0 + (0.25 * (math.pow(U_m, 2.0) / math.pow(J_m, 2.0))))))
	else:
		tmp = t_0
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(cos(Float64(0.5 * K)) * Float64(-2.0 * J_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(J_m * Float64(U_m * sqrt(Float64(0.25 / (J_m ^ 2.0))))));
	elseif (t_2 <= -2e+167)
		tmp = t_0;
	elseif (t_2 <= -1e-145)
		tmp = Float64(-2.0 * Float64(J_m * sqrt(Float64(1.0 + Float64(0.25 * Float64((U_m ^ 2.0) / (J_m ^ 2.0)))))));
	else
		tmp = t_0;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((0.5 * K)) * (-2.0 * J_m);
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (J_m * (U_m * sqrt((0.25 / (J_m ^ 2.0)))));
	elseif (t_2 <= -2e+167)
		tmp = t_0;
	elseif (t_2 <= -1e-145)
		tmp = -2.0 * (J_m * sqrt((1.0 + (0.25 * ((U_m ^ 2.0) / (J_m ^ 2.0))))));
	else
		tmp = t_0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(J$95$m * N[(U$95$m * N[Sqrt[N[(0.25 / N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+167], t$95$0, If[LessEqual[t$95$2, -1e-145], N[(-2.0 * N[(J$95$m * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[U$95$m, 2.0], $MachinePrecision] / N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.2%

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

   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 24.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 18.8

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

   7. Applied rewrites 18.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e167 or -9.99999999999999915e-146 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

   3. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in J around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

   if -2.0000000000000001e167 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999915e-146

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 31.5

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

   4. Applied rewrites 31.5%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 55.3% accurate, 0.7× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
         (- INFINITY))
      (* -2.0 (* J_m (* U_m (sqrt (/ 0.25 (pow J_m 2.0))))))
      (* (cos (* 0.5 K)) (* -2.0 J_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -((double) INFINITY)) {
		tmp = -2.0 * (J_m * (U_m * sqrt((0.25 / pow(J_m, 2.0)))));
	} else {
		tmp = cos((0.5 * K)) * (-2.0 * J_m);
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (J_m * (U_m * Math.sqrt((0.25 / Math.pow(J_m, 2.0)))));
	} else {
		tmp = Math.cos((0.5 * K)) * (-2.0 * J_m);
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -math.inf:
		tmp = -2.0 * (J_m * (U_m * math.sqrt((0.25 / math.pow(J_m, 2.0)))))
	else:
		tmp = math.cos((0.5 * K)) * (-2.0 * J_m)
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(J_m * Float64(U_m * sqrt(Float64(0.25 / (J_m ^ 2.0))))));
	else
		tmp = Float64(cos(Float64(0.5 * K)) * Float64(-2.0 * J_m));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)))) <= -Inf)
		tmp = -2.0 * (J_m * (U_m * sqrt((0.25 / (J_m ^ 2.0)))));
	else
		tmp = cos((0.5 * K)) * (-2.0 * J_m);
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(-2.0 * N[(J$95$m * N[(U$95$m * N[Sqrt[N[(0.25 / N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.2%

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

   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 24.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 18.8

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

   7. Applied rewrites 18.8%

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

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

   1. Initial program 73.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. cosh-asinh-rev N/A

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

      8. lower-cosh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. *-commutative N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

   3. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in J around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

Alternative 13: 34.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;U\_m \leq 1.25 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, \left(\left(0.25 \cdot J\_m\right) \cdot K\right) \cdot K\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J\_m \cdot \left(U\_m \cdot \sqrt{\frac{0.25}{{J\_m}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= U_m 1.25e+25)
    (* (fma -2.0 J_m (* (* (* 0.25 J_m) K) K)) 1.0)
    (* -2.0 (* J_m (* U_m (sqrt (/ 0.25 (pow J_m 2.0)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (U_m <= 1.25e+25) {
		tmp = fma(-2.0, J_m, (((0.25 * J_m) * K) * K)) * 1.0;
	} else {
		tmp = -2.0 * (J_m * (U_m * sqrt((0.25 / pow(J_m, 2.0)))));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (U_m <= 1.25e+25)
		tmp = Float64(fma(-2.0, J_m, Float64(Float64(Float64(0.25 * J_m) * K) * K)) * 1.0);
	else
		tmp = Float64(-2.0 * Float64(J_m * Float64(U_m * sqrt(Float64(0.25 / (J_m ^ 2.0))))));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[U$95$m, 1.25e+25], N[(N[(-2.0 * J$95$m + N[(N[(N[(0.25 * J$95$m), $MachinePrecision] * K), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(-2.0 * N[(J$95$m * N[(U$95$m * N[Sqrt[N[(0.25 / N[Power[J$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if U < 1.25000000000000006e25

   1. Initial program 73.2%

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

   2. Taylor expanded in J around inf

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 26.6

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

   7. Applied rewrites 26.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 26.7

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

   9. Applied rewrites 26.7%

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

   if 1.25000000000000006e25 < U

   1. Initial program 73.2%

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

   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 24.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 18.8

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

   7. Applied rewrites 18.8%

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

Alternative 14: 26.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(\mathsf{fma}\left(-2, J\_m, \left(\left(0.25 \cdot J\_m\right) \cdot K\right) \cdot K\right) \cdot 1\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (* (fma -2.0 J_m (* (* (* 0.25 J_m) K) K)) 1.0)))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (fma(-2.0, J_m, (((0.25 * J_m) * K) * K)) * 1.0);
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(fma(-2.0, J_m, Float64(Float64(Float64(0.25 * J_m) * K) * K)) * 1.0))
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * N[(N[(-2.0 * J$95$m + N[(N[(N[(0.25 * J$95$m), $MachinePrecision] * K), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.2%

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

2. Taylor expanded in J around inf

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

4. Applied rewrites 51.4%

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

5. Taylor expanded in K around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 26.6

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

7. Applied rewrites 26.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 26.7

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

9. Applied rewrites 26.7%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))