Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.5% → 99.6%

Time: 5.9s

Alternatives: 12

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: 73.5% 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: 99.6% accurate, 0.3× speedup

Math

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

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 (* 0.5 K)))
        (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))
      (- U_m)
      (if (<= t_2 2e+301)
        (*
         (* (* -2.0 J_m) t_0)
         (sqrt (+ (pow (/ U_m (* (+ J_m J_m) t_0)) 2.0) 1.0)))
        (* (* -0.5 U_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((0.5 * K));
	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 = -U_m;
	} else if (t_2 <= 2e+301) {
		tmp = ((-2.0 * J_m) * t_0) * sqrt((pow((U_m / ((J_m + J_m) * t_0)), 2.0) + 1.0));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 2e+301) {
		tmp = ((-2.0 * J_m) * t_0) * Math.sqrt((Math.pow((U_m / ((J_m + J_m) * t_0)), 2.0) + 1.0));
	} else {
		tmp = (-0.5 * U_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((0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 2e+301:
		tmp = ((-2.0 * J_m) * t_0) * math.sqrt((math.pow((U_m / ((J_m + J_m) * t_0)), 2.0) + 1.0))
	else:
		tmp = (-0.5 * U_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(0.5 * K))
	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(-U_m);
	elseif (t_2 <= 2e+301)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64((Float64(U_m / Float64(Float64(J_m + J_m) * t_0)) ^ 2.0) + 1.0)));
	else
		tmp = Float64(Float64(-0.5 * U_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((0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 2e+301)
		tmp = ((-2.0 * J_m) * t_0) * sqrt((((U_m / ((J_m + J_m) * t_0)) ^ 2.0) + 1.0));
	else
		tmp = (-0.5 * U_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[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * 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)], (-U$95$m), If[LessEqual[t$95$2, 2e+301], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(J$95$m + J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   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))))) < 2.00000000000000011e301

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 73.5

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

   4. Applied rewrites 73.5%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 73.5

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

   7. Applied rewrites 73.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 73.5

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 73.5

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

   9. Applied rewrites 73.5%

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

   if 2.00000000000000011e301 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 14.8

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

   10. Applied rewrites 14.8%

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

Alternative 2: 98.0% accurate, 0.3× speedup

Math

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

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 (* 0.5 K)))
        (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))
      (- U_m)
      (if (<= t_2 2e+301)
        (* (* (* -2.0 J_m) t_0) (cosh (asinh (/ U_m (* (+ J_m J_m) t_0)))))
        (* (* -0.5 U_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((0.5 * K));
	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 = -U_m;
	} else if (t_2 <= 2e+301) {
		tmp = ((-2.0 * J_m) * t_0) * cosh(asinh((U_m / ((J_m + J_m) * t_0))));
	} else {
		tmp = (-0.5 * U_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((0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 2e+301:
		tmp = ((-2.0 * J_m) * t_0) * math.cosh(math.asinh((U_m / ((J_m + J_m) * t_0))))
	else:
		tmp = (-0.5 * U_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(0.5 * K))
	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(-U_m);
	elseif (t_2 <= 2e+301)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * cosh(asinh(Float64(U_m / Float64(Float64(J_m + J_m) * t_0)))));
	else
		tmp = Float64(Float64(-0.5 * U_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((0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 2e+301)
		tmp = ((-2.0 * J_m) * t_0) * cosh(asinh((U_m / ((J_m + J_m) * t_0))));
	else
		tmp = (-0.5 * U_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[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * 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)], (-U$95$m), If[LessEqual[t$95$2, 2e+301], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U$95$m / N[(N[(J$95$m + J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   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))))) < 2.00000000000000011e301

   1. Initial program 73.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. cosh-asinh-rev N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-asinh.f64 N/A

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in K around 0

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

      1. lift-*.f64 84.6

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

   6. Applied rewrites 84.6%

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

   7. Taylor expanded in K around 0

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

      1. lift-*.f64 84.6

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

   9. Applied rewrites 84.6%

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

   if 2.00000000000000011e301 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 14.8

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

   10. Applied rewrites 14.8%

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

Alternative 3: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{J\_m} \cdot 0.5\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J\_m}{\left(U\_m \cdot U\_m\right) \cdot \frac{-0.5}{J\_m}}, J\_m \cdot \frac{-0.5}{J\_m}\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 5e+287)
        (* t_1 (sqrt (+ 1.0 (pow (* (/ U_m J_m) 0.5) 2.0))))
        (*
         (*
          U_m
          (fma 0.5 (/ J_m (* (* U_m U_m) (/ -0.5 J_m))) (* J_m (/ -0.5 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 t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+287) {
		tmp = t_1 * sqrt((1.0 + pow(((U_m / J_m) * 0.5), 2.0)));
	} else {
		tmp = (U_m * fma(0.5, (J_m / ((U_m * U_m) * (-0.5 / J_m))), (J_m * (-0.5 / 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))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+287)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(Float64(U_m / J_m) * 0.5) ^ 2.0))));
	else
		tmp = Float64(Float64(U_m * fma(0.5, Float64(J_m / Float64(Float64(U_m * U_m) * Float64(-0.5 / J_m))), Float64(J_m * Float64(-0.5 / 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 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]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+287], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(N[(U$95$m / J$95$m), $MachinePrecision] * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(U$95$m * N[(0.5 * N[(J$95$m / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(-0.5 / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J$95$m * N[(-0.5 / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $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 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   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))))) < 5e287

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.3

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

   4. Applied rewrites 64.3%

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

   if 5e287 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-div N/A

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

   7. Applied rewrites 20.9%

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

   8. Taylor expanded in J around -inf

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

      1. lower-/.f64 19.8

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

   10. Applied rewrites 19.8%

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

   11. Taylor expanded in J around -inf

       \[\leadsto \left(U \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{J}{\left(U \cdot U\right) \cdot \frac{\frac{-1}{2}}{J}}, J \cdot \frac{\frac{-1}{2}}{J}\right)\right) \cdot -2 \]
   12. Step-by-step derivation

       1. lower-/.f64 13.5

          \[\leadsto \left(U \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U \cdot U\right) \cdot \frac{-0.5}{J}}, J \cdot \frac{-0.5}{J}\right)\right) \cdot -2 \]

   13. Applied rewrites 13.5%

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

Alternative 4: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;t\_1 \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J\_m}{\left(U\_m \cdot U\_m\right) \cdot \frac{-0.5}{J\_m}}, J\_m \cdot \frac{-0.5}{J\_m}\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 5e+287)
        (* t_1 (cosh (asinh (* 0.5 (/ U_m J_m)))))
        (*
         (*
          U_m
          (fma 0.5 (/ J_m (* (* U_m U_m) (/ -0.5 J_m))) (* J_m (/ -0.5 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 t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+287) {
		tmp = t_1 * cosh(asinh((0.5 * (U_m / J_m))));
	} else {
		tmp = (U_m * fma(0.5, (J_m / ((U_m * U_m) * (-0.5 / J_m))), (J_m * (-0.5 / 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))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+287)
		tmp = Float64(t_1 * cosh(asinh(Float64(0.5 * Float64(U_m / J_m)))));
	else
		tmp = Float64(Float64(U_m * fma(0.5, Float64(J_m / Float64(Float64(U_m * U_m) * Float64(-0.5 / J_m))), Float64(J_m * Float64(-0.5 / 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 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]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+287], N[(t$95$1 * N[Cosh[N[ArcSinh[N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(U$95$m * N[(0.5 * N[(J$95$m / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(-0.5 / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J$95$m * N[(-0.5 / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $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 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   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))))) < 5e287

   1. Initial program 73.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. cosh-asinh-rev N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-asinh.f64 N/A

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 70.7

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

   6. Applied rewrites 70.7%

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

   if 5e287 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-div N/A

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

   7. Applied rewrites 20.9%

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

   8. Taylor expanded in J around -inf

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

      1. lower-/.f64 19.8

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

   10. Applied rewrites 19.8%

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

   11. Taylor expanded in J around -inf

       \[\leadsto \left(U \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{J}{\left(U \cdot U\right) \cdot \frac{\frac{-1}{2}}{J}}, J \cdot \frac{\frac{-1}{2}}{J}\right)\right) \cdot -2 \]
   12. Step-by-step derivation

       1. lower-/.f64 13.5

          \[\leadsto \left(U \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U \cdot U\right) \cdot \frac{-0.5}{J}}, J \cdot \frac{-0.5}{J}\right)\right) \cdot -2 \]

   13. Applied rewrites 13.5%

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

Alternative 5: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+232}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-90}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J\_m}{\left(U\_m \cdot U\_m\right) \cdot \frac{-0.5}{J\_m}}, J\_m \cdot \frac{-0.5}{J\_m}\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_3
         (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 -2e+232)
        t_3
        (if (<= t_2 -5e-90)
          (* t_1 (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)))
          (if (<= t_2 -2e-277)
            t_3
            (if (<= t_2 5e+287)
              (* (* (* -2.0 J_m) (cos (* 0.5 K))) 1.0)
              (*
               (*
                U_m
                (fma
                 0.5
                 (/ J_m (* (* U_m U_m) (/ -0.5 J_m)))
                 (* J_m (/ -0.5 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 t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_3 = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -2e+232) {
		tmp = t_3;
	} else if (t_2 <= -5e-90) {
		tmp = t_1 * sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0));
	} else if (t_2 <= -2e-277) {
		tmp = t_3;
	} else if (t_2 <= 5e+287) {
		tmp = ((-2.0 * J_m) * cos((0.5 * K))) * 1.0;
	} else {
		tmp = (U_m * fma(0.5, (J_m / ((U_m * U_m) * (-0.5 / J_m))), (J_m * (-0.5 / 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))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_3 = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -2e+232)
		tmp = t_3;
	elseif (t_2 <= -5e-90)
		tmp = Float64(t_1 * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)));
	elseif (t_2 <= -2e-277)
		tmp = t_3;
	elseif (t_2 <= 5e+287)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K))) * 1.0);
	else
		tmp = Float64(Float64(U_m * fma(0.5, Float64(J_m / Float64(Float64(U_m * U_m) * Float64(-0.5 / J_m))), Float64(J_m * Float64(-0.5 / 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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 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$3 = N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -2e+232], t$95$3, If[LessEqual[t$95$2, -5e-90], N[(t$95$1 * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-277], t$95$3, If[LessEqual[t$95$2, 5e+287], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(U$95$m * N[(0.5 * N[(J$95$m / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(-0.5 / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J$95$m * N[(-0.5 / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 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 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   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))))) < -2.00000000000000011e232 or -5.00000000000000019e-90 < (*.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))))) < -1.99999999999999994e-277

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 45.2

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

   6. Applied rewrites 45.2%

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

   if -2.00000000000000011e232 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000019e-90

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 50.4

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

   4. Applied rewrites 50.4%

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

   if -1.99999999999999994e-277 < (*.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))))) < 5e287

   1. Initial program 73.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. cosh-asinh-rev N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-asinh.f64 N/A

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in K around 0

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

      1. lift-*.f64 84.6

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

   6. Applied rewrites 84.6%

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

   7. Taylor expanded in K around 0

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

      1. lift-*.f64 84.6

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

   9. Applied rewrites 84.6%

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

   10. Taylor expanded in J around inf

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

   12. Applied rewrites 51.7%

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

   if 5e287 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-div N/A

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

   7. Applied rewrites 20.9%

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

   8. Taylor expanded in J around -inf

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

      1. lower-/.f64 19.8

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

   10. Applied rewrites 19.8%

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

   11. Taylor expanded in J around -inf

       \[\leadsto \left(U \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{J}{\left(U \cdot U\right) \cdot \frac{\frac{-1}{2}}{J}}, J \cdot \frac{\frac{-1}{2}}{J}\right)\right) \cdot -2 \]
   12. Step-by-step derivation

       1. lower-/.f64 13.5

          \[\leadsto \left(U \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U \cdot U\right) \cdot \frac{-0.5}{J}}, J \cdot \frac{-0.5}{J}\right)\right) \cdot -2 \]

   13. Applied rewrites 13.5%

       \[\leadsto \left(U \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U \cdot U\right) \cdot \frac{-0.5}{J}}, J \cdot \frac{-0.5}{J}\right)\right) \cdot -2 \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 1\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+128}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(U\_m \cdot \mathsf{fma}\left(0.5, \frac{J\_m}{\left(U\_m \cdot U\_m\right) \cdot \frac{-0.5}{J\_m}}, J\_m \cdot \frac{-0.5}{J\_m}\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (* (* (* -2.0 J_m) (cos (* 0.5 K))) 1.0)))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -1e+128)
        t_2
        (if (<= t_1 -2e-277)
          (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
          (if (<= t_1 5e+287)
            t_2
            (*
             (*
              U_m
              (fma
               0.5
               (/ J_m (* (* U_m U_m) (/ -0.5 J_m)))
               (* J_m (/ -0.5 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 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 = ((-2.0 * J_m) * cos((0.5 * K))) * 1.0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e+128) {
		tmp = t_2;
	} else if (t_1 <= -2e-277) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_1 <= 5e+287) {
		tmp = t_2;
	} else {
		tmp = (U_m * fma(0.5, (J_m / ((U_m * U_m) * (-0.5 / J_m))), (J_m * (-0.5 / 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))
	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 = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K))) * 1.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+128)
		tmp = t_2;
	elseif (t_1 <= -2e-277)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0);
	elseif (t_1 <= 5e+287)
		tmp = t_2;
	else
		tmp = Float64(Float64(U_m * fma(0.5, Float64(J_m / Float64(Float64(U_m * U_m) * Float64(-0.5 / J_m))), Float64(J_m * Float64(-0.5 / 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[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[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e+128], t$95$2, If[LessEqual[t$95$1, -2e-277], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 5e+287], t$95$2, N[(N[(U$95$m * N[(0.5 * N[(J$95$m / N[(N[(U$95$m * U$95$m), $MachinePrecision] * N[(-0.5 / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J$95$m * N[(-0.5 / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   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))))) < -1.0000000000000001e128 or -1.99999999999999994e-277 < (*.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))))) < 5e287

   1. Initial program 73.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. cosh-asinh-rev N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-asinh.f64 N/A

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in K around 0

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

      1. lift-*.f64 84.6

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

   6. Applied rewrites 84.6%

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

   7. Taylor expanded in K around 0

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

      1. lift-*.f64 84.6

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

   9. Applied rewrites 84.6%

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

   10. Taylor expanded in J around inf

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

   12. Applied rewrites 51.7%

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

   if -1.0000000000000001e128 < (*.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))))) < -1.99999999999999994e-277

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 45.2

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

   6. Applied rewrites 45.2%

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

   if 5e287 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. sqrt-div N/A

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

   7. Applied rewrites 20.9%

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

   8. Taylor expanded in J around -inf

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

      1. lower-/.f64 19.8

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

   10. Applied rewrites 19.8%

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

   11. Taylor expanded in J around -inf

       \[\leadsto \left(U \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{J}{\left(U \cdot U\right) \cdot \frac{\frac{-1}{2}}{J}}, J \cdot \frac{\frac{-1}{2}}{J}\right)\right) \cdot -2 \]
   12. Step-by-step derivation

       1. lower-/.f64 13.5

          \[\leadsto \left(U \cdot \mathsf{fma}\left(0.5, \frac{J}{\left(U \cdot U\right) \cdot \frac{-0.5}{J}}, J \cdot \frac{-0.5}{J}\right)\right) \cdot -2 \]

   13. Applied rewrites 13.5%

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

Alternative 7: 77.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -2e-277)
        (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
        (* (* -0.5 U_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 t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-277) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else {
		tmp = (-0.5 * U_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))
	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))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-277)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_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[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]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-277], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   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))))) < -1.99999999999999994e-277

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 45.2

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

   6. Applied rewrites 45.2%

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

   if -1.99999999999999994e-277 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 14.8

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

   10. Applied rewrites 14.8%

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

Alternative 8: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-162}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -5e-162)
        (* (* (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)) J_m) -2.0)
        (* (* -0.5 U_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 t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e-162) {
		tmp = (sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else {
		tmp = (-0.5 * U_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))
	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))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-162)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)) * J_m) * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_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[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]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -5e-162], N[(N[(N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000014e-162

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   if -5.00000000000000014e-162 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 14.8

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

   10. Applied rewrites 14.8%

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

Alternative 9: 60.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-162}:\\ \;\;\;\;\left(\left(U\_m \cdot \frac{U\_m}{J\_m \cdot J\_m}\right) \cdot -0.25 - 2\right) \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -5e-162)
        (* (- (* (* U_m (/ U_m (* J_m J_m))) -0.25) 2.0) J_m)
        (* (* -0.5 U_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 t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e-162) {
		tmp = (((U_m * (U_m / (J_m * J_m))) * -0.25) - 2.0) * J_m;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= -5e-162) {
		tmp = (((U_m * (U_m / (J_m * J_m))) * -0.25) - 2.0) * J_m;
	} else {
		tmp = (-0.5 * U_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))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= -5e-162:
		tmp = (((U_m * (U_m / (J_m * J_m))) * -0.25) - 2.0) * J_m
	else:
		tmp = (-0.5 * U_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))
	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))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-162)
		tmp = Float64(Float64(Float64(Float64(U_m * Float64(U_m / Float64(J_m * J_m))) * -0.25) - 2.0) * J_m);
	else
		tmp = Float64(Float64(-0.5 * U_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));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= -5e-162)
		tmp = (((U_m * (U_m / (J_m * J_m))) * -0.25) - 2.0) * J_m;
	else
		tmp = (-0.5 * U_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]}, 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]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -5e-162], N[(N[(N[(N[(U$95$m * N[(U$95$m / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] - 2.0), $MachinePrecision] * J$95$m), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000014e-162

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 18.3

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

   7. Applied rewrites 18.3%

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

   8. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 27.9

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

   10. Applied rewrites 27.9%

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

   if -5.00000000000000014e-162 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 14.8

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

   10. Applied rewrites 14.8%

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

Alternative 10: 60.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, -0.25 \cdot \frac{U\_m \cdot U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -2e-277)
        (fma -2.0 J_m (* -0.25 (/ (* U_m U_m) J_m)))
        (* (* -0.5 U_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 t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-277) {
		tmp = fma(-2.0, J_m, (-0.25 * ((U_m * U_m) / J_m)));
	} else {
		tmp = (-0.5 * U_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))
	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))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-277)
		tmp = fma(-2.0, J_m, Float64(-0.25 * Float64(Float64(U_m * U_m) / J_m)));
	else
		tmp = Float64(Float64(-0.5 * U_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[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]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-277], N[(-2.0 * J$95$m + N[(-0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $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 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]

   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))))) < -1.99999999999999994e-277

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in U around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 28.2

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

   7. Applied rewrites 28.2%

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

   if -1.99999999999999994e-277 < (*.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 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 14.8

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

   10. Applied rewrites 14.8%

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

Alternative 11: 52.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

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)) -5e-310) (* (* -0.5 U_m) -2.0) (- U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (cos((K / 2.0)) <= -5e-310) {
		tmp = (-0.5 * U_m) * -2.0;
	} else {
		tmp = -U_m;
	}
	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)) <= (-5d-310)) then
        tmp = ((-0.5d0) * u_m) * (-2.0d0)
    else
        tmp = -u_m
    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)) <= -5e-310) {
		tmp = (-0.5 * U_m) * -2.0;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if math.cos((K / 2.0)) <= -5e-310:
		tmp = (-0.5 * U_m) * -2.0
	else:
		tmp = -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -5e-310)
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -5e-310)
		tmp = (-0.5 * U_m) * -2.0;
	else
		tmp = -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -5e-310], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], (-U$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -4.999999999999985e-310

   1. Initial program 73.5%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 32.1%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 14.8

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

   10. Applied rewrites 14.8%

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

   if -4.999999999999985e-310 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 73.5%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

   4. Applied rewrites 1.4%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 51.8

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in K around 0

      \[\leadsto -1 \cdot U \]
   9. Step-by-step derivation

   10. Applied rewrites 38.7%

       \[\leadsto -1 \cdot U \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot U \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(U\right) \]

       3. lower-neg.f64 38.7

          \[\leadsto -U \]

   12. Applied rewrites 38.7%

       \[\leadsto -U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 38.7% accurate, 55.9× speedup

Math

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

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 (- U_m)))

C

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

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 * -u_m
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 * -U_m;
}

Python

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

Julia

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

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * -U_m;
end

Wolfram

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

Derivation

1. Initial program 73.5%

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

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. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

4. Applied rewrites 1.4%

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

5. Taylor expanded in J around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 51.8

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

7. Applied rewrites 51.8%

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

8. Taylor expanded in K around 0

   \[\leadsto -1 \cdot U \]
9. Step-by-step derivation

10. Applied rewrites 38.7%

    \[\leadsto -1 \cdot U \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto -1 \cdot U \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(U\right) \]

    3. lower-neg.f64 38.7

       \[\leadsto -U \]

12. Applied rewrites 38.7%

    \[\leadsto -U \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025138 
(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)))))