Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 99.4%

Time: 6.0s

Alternatives: 11

Speedup: 0.4×

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: 73.3% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \left(-1 \cdot \frac{t\_1}{\left|t\_1\right|}\right) \cdot U\_m\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_4 := \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_4\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_4}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 (cos (* -0.5 K)))
        (t_2 (* (* -1.0 (/ t_1 (fabs t_1))) U_m))
        (t_3
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_4 (cos (* K 0.5))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      t_2
      (if (<= t_3 2e+300)
        (*
         (* (* -2.0 J_m) t_4)
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_4)) 2.0))))
        t_2)))))

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 = cos((-0.5 * K));
	double t_2 = (-1.0 * (t_1 / fabs(t_1))) * U_m;
	double t_3 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_4 = cos((K * 0.5));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= 2e+300) {
		tmp = ((-2.0 * J_m) * t_4) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_4)), 2.0)));
	} else {
		tmp = t_2;
	}
	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 = Math.cos((-0.5 * K));
	double t_2 = (-1.0 * (t_1 / Math.abs(t_1))) * U_m;
	double t_3 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_4 = Math.cos((K * 0.5));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_3 <= 2e+300) {
		tmp = ((-2.0 * J_m) * t_4) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_4)), 2.0)));
	} else {
		tmp = t_2;
	}
	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 = math.cos((-0.5 * K))
	t_2 = (-1.0 * (t_1 / math.fabs(t_1))) * U_m
	t_3 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	t_4 = math.cos((K * 0.5))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_2
	elif t_3 <= 2e+300:
		tmp = ((-2.0 * J_m) * t_4) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_4)), 2.0)))
	else:
		tmp = t_2
	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 = cos(Float64(-0.5 * K))
	t_2 = Float64(Float64(-1.0 * Float64(t_1 / abs(t_1))) * U_m)
	t_3 = 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_4 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= 2e+300)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_4) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_4)) ^ 2.0))));
	else
		tmp = t_2;
	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 = cos((-0.5 * K));
	t_2 = (-1.0 * (t_1 / abs(t_1))) * U_m;
	t_3 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	t_4 = cos((K * 0.5));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_2;
	elseif (t_3 <= 2e+300)
		tmp = ((-2.0 * J_m) * t_4) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_4)) ^ 2.0)));
	else
		tmp = t_2;
	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[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 * N[(t$95$1 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]}, Block[{t$95$3 = 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$4 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, (-Infinity)], t$95$2, If[LessEqual[t$95$3, 2e+300], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$4), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.0000000000000001e300 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 1.4%

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

   6. Applied rewrites 1.4%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 52.0

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

   9. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e300

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.3

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

   3. Applied rewrites 73.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ t_3 := \left(-1 \cdot \frac{t\_2}{\left|t\_2\right|}\right) \cdot U\_m\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(t\_2 \cdot -2\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J\_m}, \frac{U\_m}{J\_m} \cdot 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

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 (* (* -1.0 (/ t_2 (fabs t_2))) U_m)))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      t_3
      (if (<= t_1 2e+300)
        (*
         (* (* t_2 -2.0) J_m)
         (sqrt
          (fma
           (/ U_m (* (fma (cos K) 0.5 0.5) J_m))
           (* (/ U_m J_m) 0.25)
           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 = (-1.0 * (t_2 / fabs(t_2))) * U_m;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 2e+300) {
		tmp = ((t_2 * -2.0) * J_m) * sqrt(fma((U_m / (fma(cos(K), 0.5, 0.5) * J_m)), ((U_m / J_m) * 0.25), 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(Float64(-1.0 * Float64(t_2 / abs(t_2))) * U_m)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+300)
		tmp = Float64(Float64(Float64(t_2 * -2.0) * J_m) * sqrt(fma(Float64(U_m / Float64(fma(cos(K), 0.5, 0.5) * J_m)), Float64(Float64(U_m / J_m) * 0.25), 1.0)));
	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[(N[(-1.0 * N[(t$95$2 / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+300], N[(N[(N[(t$95$2 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(U$95$m / J$95$m), $MachinePrecision] * 0.25), $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 2.0000000000000001e300 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 1.4%

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

   6. Applied rewrites 1.4%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 52.0

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

   9. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e300

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   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 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 3: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ t_3 := \left(t\_2 \cdot -2\right) \cdot J\_m\\ t_4 := \left(-1 \cdot \frac{t\_2}{\left|t\_2\right|}\right) \cdot U\_m\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-86}:\\ \;\;\;\;t\_3 \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m}, 0.25 \cdot \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+265}:\\ \;\;\;\;t\_3 \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot \left(\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot J\_m\right)}, 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \end{array} \]

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 (* (* t_2 -2.0) J_m))
        (t_4 (* (* -1.0 (/ t_2 (fabs t_2))) U_m)))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      t_4
      (if (<= t_1 5e-86)
        (* t_3 (sqrt (fma (/ U_m J_m) (* 0.25 (/ U_m J_m)) 1.0)))
        (if (<= t_1 2e+265)
          (*
           t_3
           (sqrt
            (fma
             (/ (* U_m U_m) (* J_m (* (fma (cos K) 0.5 0.5) J_m)))
             0.25
             1.0)))
          t_4))))))

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 = (t_2 * -2.0) * J_m;
	double t_4 = (-1.0 * (t_2 / fabs(t_2))) * U_m;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_1 <= 5e-86) {
		tmp = t_3 * sqrt(fma((U_m / J_m), (0.25 * (U_m / J_m)), 1.0));
	} else if (t_1 <= 2e+265) {
		tmp = t_3 * sqrt(fma(((U_m * U_m) / (J_m * (fma(cos(K), 0.5, 0.5) * J_m))), 0.25, 1.0));
	} else {
		tmp = t_4;
	}
	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(Float64(t_2 * -2.0) * J_m)
	t_4 = Float64(Float64(-1.0 * Float64(t_2 / abs(t_2))) * U_m)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_1 <= 5e-86)
		tmp = Float64(t_3 * sqrt(fma(Float64(U_m / J_m), Float64(0.25 * Float64(U_m / J_m)), 1.0)));
	elseif (t_1 <= 2e+265)
		tmp = Float64(t_3 * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * Float64(fma(cos(K), 0.5, 0.5) * J_m))), 0.25, 1.0)));
	else
		tmp = t_4;
	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[(N[(t$95$2 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 * N[(t$95$2 / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$4, If[LessEqual[t$95$1, 5e-86], N[(t$95$3 * N[Sqrt[N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(0.25 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+265], N[(t$95$3 * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$4]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 2.00000000000000013e265 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 1.4%

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

   6. Applied rewrites 1.4%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 52.0

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

   9. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999999e-86

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   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 73.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 64.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 64.1

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

   10. Applied rewrites 64.1%

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

   if 4.9999999999999999e-86 < (*.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))))) < 2.00000000000000013e265

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   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 73.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 73.1

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/l/ N/A

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

      14. frac-times N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 54.7

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

   7. Applied rewrites 54.7%

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

Alternative 4: 90.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ t_3 := \left(-1 \cdot \frac{t\_2}{\left|t\_2\right|}\right) \cdot U\_m\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(t\_2 \cdot -2\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m}, 0.25 \cdot \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

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 (* (* -1.0 (/ t_2 (fabs t_2))) U_m)))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      t_3
      (if (<= t_1 2e+300)
        (*
         (* (* t_2 -2.0) J_m)
         (sqrt (fma (/ U_m J_m) (* 0.25 (/ U_m J_m)) 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 = (-1.0 * (t_2 / fabs(t_2))) * U_m;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 2e+300) {
		tmp = ((t_2 * -2.0) * J_m) * sqrt(fma((U_m / J_m), (0.25 * (U_m / J_m)), 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(Float64(-1.0 * Float64(t_2 / abs(t_2))) * U_m)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+300)
		tmp = Float64(Float64(Float64(t_2 * -2.0) * J_m) * sqrt(fma(Float64(U_m / J_m), Float64(0.25 * Float64(U_m / J_m)), 1.0)));
	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[(N[(-1.0 * N[(t$95$2 / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U$95$m), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+300], N[(N[(N[(t$95$2 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(0.25 * N[(U$95$m / J$95$m), $MachinePrecision]), $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 2.0000000000000001e300 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 1.4%

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

   6. Applied rewrites 1.4%

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

   7. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 52.0

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

   9. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.0000000000000001e300

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   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 73.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 64.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 64.1

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

   10. Applied rewrites 64.1%

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

Alternative 5: 72.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(K \cdot K\right) \cdot J\_m\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\cosh \sinh^{-1} \left(\frac{U\_m}{\mathsf{fma}\left(-0.25, t\_0, J\_m + J\_m\right)}\right) \cdot \mathsf{fma}\left(J\_m, -2, t\_0 \cdot 0.25\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m}, 0.25 \cdot \frac{U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(J\_m \cdot \left(U\_m \cdot \sqrt{\frac{0.25}{{J\_m}^{2}}}\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (* K K) 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))
      (*
       (cosh (asinh (/ U_m (fma -0.25 t_0 (+ J_m J_m)))))
       (fma J_m -2.0 (* t_0 0.25)))
      (if (<= t_2 2e+300)
        (*
         (* (* (cos (* -0.5 K)) -2.0) J_m)
         (sqrt (fma (/ U_m J_m) (* 0.25 (/ U_m J_m)) 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 t_0 = (K * K) * 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 = cosh(asinh((U_m / fma(-0.25, t_0, (J_m + J_m))))) * fma(J_m, -2.0, (t_0 * 0.25));
	} else if (t_2 <= 2e+300) {
		tmp = ((cos((-0.5 * K)) * -2.0) * J_m) * sqrt(fma((U_m / J_m), (0.25 * (U_m / J_m)), 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)
	t_0 = Float64(Float64(K * K) * 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(cosh(asinh(Float64(U_m / fma(-0.25, t_0, Float64(J_m + J_m))))) * fma(J_m, -2.0, Float64(t_0 * 0.25)));
	elseif (t_2 <= 2e+300)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * -2.0) * J_m) * sqrt(fma(Float64(U_m / J_m), Float64(0.25 * Float64(U_m / J_m)), 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_] := Block[{t$95$0 = N[(N[(K * K), $MachinePrecision] * J$95$m), $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[(N[Cosh[N[ArcSinh[N[(U$95$m / N[(-0.25 * t$95$0 + N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0 + N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+300], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(0.25 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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 3 regimes

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

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 62.3

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

   4. Applied rewrites 62.3%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 38.8

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

   7. Applied rewrites 38.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 38.8

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

   9. Applied rewrites 45.2%

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

   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))))) < 2.0000000000000001e300

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   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 73.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.1%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 64.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 64.1

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

   10. Applied rewrites 64.1%

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

   if 2.0000000000000001e300 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 7.3

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

   7. Applied rewrites 7.3%

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

Alternative 6: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(K \cdot K\right) \cdot J\_m\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;K \leq 52000000:\\ \;\;\;\;\cosh \sinh^{-1} \left(\frac{U\_m}{\mathsf{fma}\left(-0.25, t\_0, J\_m + J\_m\right)}\right) \cdot \mathsf{fma}\left(J\_m, -2, t\_0 \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot 1\\ \end{array} \end{array} \end{array} \]

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 (* (* K K) J_m)))
   (*
    J_s
    (if (<= K 52000000.0)
      (*
       (cosh (asinh (/ U_m (fma -0.25 t_0 (+ J_m J_m)))))
       (fma J_m -2.0 (* t_0 0.25)))
      (* (* (* (cos (* -0.5 K)) J_m) -2.0) 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) {
	double t_0 = (K * K) * J_m;
	double tmp;
	if (K <= 52000000.0) {
		tmp = cosh(asinh((U_m / fma(-0.25, t_0, (J_m + J_m))))) * fma(J_m, -2.0, (t_0 * 0.25));
	} else {
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * 1.0;
	}
	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(Float64(K * K) * J_m)
	tmp = 0.0
	if (K <= 52000000.0)
		tmp = Float64(cosh(asinh(Float64(U_m / fma(-0.25, t_0, Float64(J_m + J_m))))) * fma(J_m, -2.0, Float64(t_0 * 0.25)));
	else
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J_m) * -2.0) * 1.0);
	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_] := Block[{t$95$0 = N[(N[(K * K), $MachinePrecision] * J$95$m), $MachinePrecision]}, N[(J$95$s * If[LessEqual[K, 52000000.0], N[(N[Cosh[N[ArcSinh[N[(U$95$m / N[(-0.25 * t$95$0 + N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0 + N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 5.2e7

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 62.3

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

   4. Applied rewrites 62.3%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 38.8

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

   7. Applied rewrites 38.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 38.8

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

   9. Applied rewrites 45.2%

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

   if 5.2e7 < K

   1. Initial program 73.3%

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

   2. Taylor expanded in J around inf

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

   4. Applied rewrites 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 51.9

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. lift-*.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 7: 56.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.975:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\frac{\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}}{4} - -1}\\ \end{array} \end{array} \]

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 (<= (cos (/ K 2.0)) 0.975)
    (* (* (* (cos (* -0.5 K)) J_m) -2.0) 1.0)
    (* (* -2.0 J_m) (sqrt (- (/ (* (/ U_m J_m) (/ U_m J_m)) 4.0) -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) {
	double tmp;
	if (cos((K / 2.0)) <= 0.975) {
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * 1.0;
	} else {
		tmp = (-2.0 * J_m) * sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0));
	}
	return J_s * tmp;
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
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_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= 0.975d0) then
        tmp = ((cos(((-0.5d0) * k)) * j_m) * (-2.0d0)) * 1.0d0
    else
        tmp = ((-2.0d0) * j_m) * sqrt(((((u_m / j_m) * (u_m / j_m)) / 4.0d0) - (-1.0d0)))
    end if
    code = j_s * tmp
end function

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 tmp;
	if (Math.cos((K / 2.0)) <= 0.975) {
		tmp = ((Math.cos((-0.5 * K)) * J_m) * -2.0) * 1.0;
	} else {
		tmp = (-2.0 * J_m) * Math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0));
	}
	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):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.975:
		tmp = ((math.cos((-0.5 * K)) * J_m) * -2.0) * 1.0
	else:
		tmp = (-2.0 * J_m) * math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0))
	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 (cos(Float64(K / 2.0)) <= 0.975)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J_m) * -2.0) * 1.0);
	else
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(Float64(Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) / 4.0) - -1.0)));
	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)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.975)
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * 1.0;
	else
		tmp = (-2.0 * J_m) * sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0));
	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_] := N[(J$95$s * If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.975], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * 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]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.974999999999999978

   1. Initial program 73.3%

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

   2. Taylor expanded in J around inf

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

   4. Applied rewrites 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 51.9

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. lift-*.f64 51.9

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

   6. Applied rewrites 51.9%

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

   if 0.974999999999999978 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   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 73.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.1%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 44.2%

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

Alternative 8: 50.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}} \leq 10^{-149}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\frac{\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}}{4} - -1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(J\_m \cdot \left(U\_m \cdot \sqrt{\frac{0.25}{{J\_m}^{2}}}\right)\right)\\ \end{array} \end{array} \end{array} \]

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))))
         1e-149)
      (* (* -2.0 J_m) (sqrt (- (/ (* (/ U_m J_m) (/ U_m J_m)) 4.0) -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 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)))) <= 1e-149) {
		tmp = (-2.0 * J_m) * sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0));
	} else {
		tmp = 2.0 * (J_m * (U_m * sqrt((0.25 / pow(J_m, 2.0)))));
	}
	return J_s * tmp;
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
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_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0)))) <= 1d-149) then
        tmp = ((-2.0d0) * j_m) * sqrt(((((u_m / j_m) * (u_m / j_m)) / 4.0d0) - (-1.0d0)))
    else
        tmp = 2.0d0 * (j_m * (u_m * sqrt((0.25d0 / (j_m ** 2.0d0)))))
    end if
    code = j_s * tmp
end function

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)))) <= 1e-149) {
		tmp = (-2.0 * J_m) * Math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0));
	} else {
		tmp = 2.0 * (J_m * (U_m * Math.sqrt((0.25 / Math.pow(J_m, 2.0)))));
	}
	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((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)))) <= 1e-149:
		tmp = (-2.0 * J_m) * math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0))
	else:
		tmp = 2.0 * (J_m * (U_m * math.sqrt((0.25 / math.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)
	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)))) <= 1e-149)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(Float64(Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) / 4.0) - -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

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)))) <= 1e-149)
		tmp = (-2.0 * J_m) * sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0));
	else
		tmp = 2.0 * (J_m * (U_m * sqrt((0.25 / (J_m ^ 2.0)))));
	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[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], 1e-149], N[(N[(-2.0 * J$95$m), $MachinePrecision] * 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]), $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 (*.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))))) < 9.99999999999999979e-150

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   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 73.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 64.1%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 44.2%

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

   if 9.99999999999999979e-150 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.3%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 7.3

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

   7. Applied rewrites 7.3%

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

Alternative 9: 44.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(\left(-2 \cdot J\_m\right) \cdot \sqrt{\frac{\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}}{4} - -1}\right) \end{array} \]

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 (* (* -2.0 J_m) (sqrt (- (/ (* (/ U_m J_m) (/ U_m J_m)) 4.0) -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 * ((-2.0 * J_m) * sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)));
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
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_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * (((-2.0d0) * j_m) * sqrt(((((u_m / j_m) * (u_m / j_m)) / 4.0d0) - (-1.0d0))))
end function

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) {
	return J_s * ((-2.0 * J_m) * Math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -1.0)));
}

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 * ((-2.0 * J_m) * math.sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -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(Float64(-2.0 * J_m) * sqrt(Float64(Float64(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)) / 4.0) - -1.0))))
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 * ((-2.0 * J_m) * sqrt(((((U_m / J_m) * (U_m / J_m)) / 4.0) - -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), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f64 N/A

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

   9. frac-times N/A

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

   10. associate-/l* N/A

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

   11. lower-fma.f64 N/A

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

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 73.1%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 64.1%

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

9. Taylor expanded in K around 0

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

11. Applied rewrites 44.2%

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

Alternative 10: 26.5% accurate, 6.2× speedup

Math

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

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 (* K K) (* 0.25 J_m) (* J_m -2.0)) 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((K * K), (0.25 * J_m), (J_m * -2.0)) * 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(Float64(K * K), Float64(0.25 * J_m), Float64(J_m * -2.0)) * 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[(N[(K * K), $MachinePrecision] * N[(0.25 * J$95$m), $MachinePrecision] + N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in J around 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 51.9%

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

5. Taylor expanded in K around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 26.5

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

7. Applied rewrites 26.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 26.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 26.5

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

9. Applied rewrites 26.5%

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

Alternative 11: 26.5% accurate, 7.4× speedup

Math

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

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 (* (* J_m (fma (* K K) 0.25 -2.0)) 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 * ((J_m * fma((K * K), 0.25, -2.0)) * 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(Float64(J_m * fma(Float64(K * K), 0.25, -2.0)) * 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[(J$95$m * N[(N[(K * K), $MachinePrecision] * 0.25 + -2.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in J around 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 51.9%

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

5. Taylor expanded in K around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 26.5

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

7. Applied rewrites 26.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 26.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 26.5

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

9. Applied rewrites 26.5%

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

    1. lift-fma.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. pow2 N/A

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

    4. lift-pow.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*r* N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. distribute-rgt-out N/A

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

    10. lower-*.f64 N/A

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

    11. lower-fma.f64 26.5

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

    12. lift-pow.f64 N/A

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

    13. pow2 N/A

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

    14. lift-*.f64 26.5

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

11. Applied rewrites 26.5%

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

Reproduce

herbie shell --seed 2025157 
(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)))))