Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.2% → 98.5%

Time: 5.9s

Alternatives: 11

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 0.5)))
        (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 0.5) -2.0)
      (if (<= t_2 2e+272)
        (*
         (* (* -2.0 J_m) t_0)
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 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 * 0.5));
	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 * 0.5) * -2.0;
	} else if (t_2 <= 2e+272) {
		tmp = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	} 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((K * 0.5));
	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 * 0.5) * -2.0;
	} else if (t_2 <= 2e+272) {
		tmp = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	} 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((K * 0.5))
	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 * 0.5) * -2.0
	elif t_2 <= 2e+272:
		tmp = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 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 * 0.5))
	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(Float64(U_m * 0.5) * -2.0);
	elseif (t_2 <= 2e+272)
		tmp = 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))));
	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 * 0.5));
	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 * 0.5) * -2.0;
	elseif (t_2 <= 2e+272)
		tmp = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	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[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+272], 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[(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 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   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))))) < 2.0000000000000001e272

   1. Initial program 73.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.2

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

   3. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if 2.0000000000000001e272 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

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

Alternative 2: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \left(t\_0 \cdot \cosh \sinh^{-1} \left(\frac{U\_m}{t\_0 \cdot \left(J\_m + J\_m\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 0.5) -2.0)
      (if (<= t_2 2e+272)
        (* (* J_m -2.0) (* t_0 (cosh (asinh (/ U_m (* t_0 (+ J_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((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 * 0.5) * -2.0;
	} else if (t_2 <= 2e+272) {
		tmp = (J_m * -2.0) * (t_0 * cosh(asinh((U_m / (t_0 * (J_m + 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((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 * 0.5) * -2.0
	elif t_2 <= 2e+272:
		tmp = (J_m * -2.0) * (t_0 * math.cosh(math.asinh((U_m / (t_0 * (J_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(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(Float64(U_m * 0.5) * -2.0);
	elseif (t_2 <= 2e+272)
		tmp = Float64(Float64(J_m * -2.0) * Float64(t_0 * cosh(asinh(Float64(U_m / Float64(t_0 * Float64(J_m + 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((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 * 0.5) * -2.0;
	elseif (t_2 <= 2e+272)
		tmp = (J_m * -2.0) * (t_0 * cosh(asinh((U_m / (t_0 * (J_m + 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[(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)], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 2e+272], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[(t$95$0 * N[Cosh[N[ArcSinh[N[(U$95$m / N[(t$95$0 * N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]), $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 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   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))))) < 2.0000000000000001e272

   1. Initial program 73.2%

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

   3. Applied rewrites 84.8%

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

   if 2.0000000000000001e272 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

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

Alternative 3: 90.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot 1\\ t_1 := \left(-0.5 \cdot U\_m\right) \cdot -2\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{+157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\left(\left(\cos \left(K \cdot 0.5\right) \cdot J\_m\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_3 \leq 10^{-175}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (* (* (cos (* 0.5 K)) J_m) -2.0) 1.0))
        (t_1 (* (* -0.5 U_m) -2.0))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0))))))
   (*
    J_s
    (if (<= t_3 -2e+304)
      (* (* U_m 0.5) -2.0)
      (if (<= t_3 -4e+157)
        t_0
        (if (<= t_3 -5e-44)
          (*
           (* (* (cos (* K 0.5)) J_m) -2.0)
           (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)))
          (if (<= t_3 -2e-263)
            (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
            (if (<= t_3 1e-175)
              t_0
              (if (<= t_3 1e+60) t_1 (if (<= t_3 2e+272) t_0 t_1))))))))))

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)) * J_m) * -2.0) * 1.0;
	double t_1 = (-0.5 * U_m) * -2.0;
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -2e+304) {
		tmp = (U_m * 0.5) * -2.0;
	} else if (t_3 <= -4e+157) {
		tmp = t_0;
	} else if (t_3 <= -5e-44) {
		tmp = ((cos((K * 0.5)) * J_m) * -2.0) * sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0));
	} else if (t_3 <= -2e-263) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_3 <= 1e-175) {
		tmp = t_0;
	} else if (t_3 <= 1e+60) {
		tmp = t_1;
	} else if (t_3 <= 2e+272) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(Float64(cos(Float64(0.5 * K)) * J_m) * -2.0) * 1.0)
	t_1 = Float64(Float64(-0.5 * U_m) * -2.0)
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -2e+304)
		tmp = Float64(Float64(U_m * 0.5) * -2.0);
	elseif (t_3 <= -4e+157)
		tmp = t_0;
	elseif (t_3 <= -5e-44)
		tmp = Float64(Float64(Float64(cos(Float64(K * 0.5)) * J_m) * -2.0) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)));
	elseif (t_3 <= -2e-263)
		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_3 <= 1e-175)
		tmp = t_0;
	elseif (t_3 <= 1e+60)
		tmp = t_1;
	elseif (t_3 <= 2e+272)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, -2e+304], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$3, -4e+157], t$95$0, If[LessEqual[t$95$3, -5e-44], N[(N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 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$3, -2e-263], 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$3, 1e-175], t$95$0, If[LessEqual[t$95$3, 1e+60], t$95$1, If[LessEqual[t$95$3, 2e+272], t$95$0, t$95$1]]]]]]]), $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))))) < -1.9999999999999999e304

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -1.9999999999999999e304 < (*.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))))) < -3.99999999999999993e157 or -2e-263 < (*.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))))) < 1e-175 or 9.9999999999999995e59 < (*.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))))) < 2.0000000000000001e272

   1. Initial program 73.2%

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in J around inf

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

   6. Applied rewrites 52.5%

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

   if -3.99999999999999993e157 < (*.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))))) < -5.00000000000000039e-44

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \]
   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 52.4

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

   4. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if -5.00000000000000039e-44 < (*.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))))) < -2e-263

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]
   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 45.6

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

   6. Applied rewrites 45.6%

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

   if 1e-175 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999995e59 or 2.0000000000000001e272 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

       \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\left(\left(\cos \left(K \cdot 0.5\right) \cdot J\_m\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 0.5) -2.0)
      (if (<= t_1 2e+272)
        (*
         (* (* (cos (* K 0.5)) J_m) -2.0)
         (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 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((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 * 0.5) * -2.0;
	} else if (t_1 <= 2e+272) {
		tmp = ((cos((K * 0.5)) * J_m) * -2.0) * sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 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(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(Float64(U_m * 0.5) * -2.0);
	elseif (t_1 <= 2e+272)
		tmp = Float64(Float64(Float64(cos(Float64(K * 0.5)) * J_m) * -2.0) * sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)));
	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)], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 2e+272], N[(N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 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]), $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 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   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))))) < 2.0000000000000001e272

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \]
   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 52.4

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

   4. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 64.8

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

   8. Applied rewrites 64.8%

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

   if 2.0000000000000001e272 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

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

Alternative 5: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot 1\\ t_1 := \left(-0.5 \cdot U\_m\right) \cdot -2\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{+205}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_3 \leq 10^{-175}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (* (* (cos (* 0.5 K)) J_m) -2.0) 1.0))
        (t_1 (* (* -0.5 U_m) -2.0))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0))))))
   (*
    J_s
    (if (<= t_3 -2e+304)
      (* (* U_m 0.5) -2.0)
      (if (<= t_3 -1e+205)
        t_0
        (if (<= t_3 -2e-263)
          (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
          (if (<= t_3 1e-175)
            t_0
            (if (<= t_3 1e+60) t_1 (if (<= t_3 2e+272) t_0 t_1)))))))))

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)) * J_m) * -2.0) * 1.0;
	double t_1 = (-0.5 * U_m) * -2.0;
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -2e+304) {
		tmp = (U_m * 0.5) * -2.0;
	} else if (t_3 <= -1e+205) {
		tmp = t_0;
	} else if (t_3 <= -2e-263) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_3 <= 1e-175) {
		tmp = t_0;
	} else if (t_3 <= 1e+60) {
		tmp = t_1;
	} else if (t_3 <= 2e+272) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(Float64(cos(Float64(0.5 * K)) * J_m) * -2.0) * 1.0)
	t_1 = Float64(Float64(-0.5 * U_m) * -2.0)
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -2e+304)
		tmp = Float64(Float64(U_m * 0.5) * -2.0);
	elseif (t_3 <= -1e+205)
		tmp = t_0;
	elseif (t_3 <= -2e-263)
		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_3 <= 1e-175)
		tmp = t_0;
	elseif (t_3 <= 1e+60)
		tmp = t_1;
	elseif (t_3 <= 2e+272)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, -2e+304], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$3, -1e+205], t$95$0, If[LessEqual[t$95$3, -2e-263], 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$3, 1e-175], t$95$0, If[LessEqual[t$95$3, 1e+60], t$95$1, If[LessEqual[t$95$3, 2e+272], t$95$0, t$95$1]]]]]]), $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))))) < -1.9999999999999999e304

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -1.9999999999999999e304 < (*.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))))) < -1.00000000000000002e205 or -2e-263 < (*.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))))) < 1e-175 or 9.9999999999999995e59 < (*.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))))) < 2.0000000000000001e272

   1. Initial program 73.2%

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in J around inf

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

   6. Applied rewrites 52.5%

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

   if -1.00000000000000002e205 < (*.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))))) < -2e-263

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]
   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 45.6

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

   6. Applied rewrites 45.6%

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

   if 1e-175 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999995e59 or 2.0000000000000001e272 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

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

Alternative 6: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 0.5) -2.0)
      (if (<= t_1 -1e-291)
        (* (* (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 * 0.5) * -2.0;
	} else if (t_1 <= -1e-291) {
		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(Float64(U_m * 0.5) * -2.0);
	elseif (t_1 <= -1e-291)
		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)], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-291], 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 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999962e-292

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]
   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 45.6

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

   6. Applied rewrites 45.6%

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

   if -9.99999999999999962e-292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

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

Alternative 7: 69.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, U\_m \cdot \frac{U\_m}{J\_m \cdot J\_m}, -2\right) \cdot J\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-93}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, -0.25 \cdot \frac{U\_m \cdot U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 -2e+304)
      (* (* U_m 0.5) -2.0)
      (if (<= t_1 -4e+157)
        (* (fma -0.25 (* U_m (/ U_m (* J_m J_m))) -2.0) J_m)
        (if (<= t_1 -5e-93)
          (* (* (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)) J_m) -2.0)
          (if (<= t_1 -1e-291)
            (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 <= -2e+304) {
		tmp = (U_m * 0.5) * -2.0;
	} else if (t_1 <= -4e+157) {
		tmp = fma(-0.25, (U_m * (U_m / (J_m * J_m))), -2.0) * J_m;
	} else if (t_1 <= -5e-93) {
		tmp = (sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_1 <= -1e-291) {
		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 <= -2e+304)
		tmp = Float64(Float64(U_m * 0.5) * -2.0);
	elseif (t_1 <= -4e+157)
		tmp = Float64(fma(-0.25, Float64(U_m * Float64(U_m / Float64(J_m * J_m))), -2.0) * J_m);
	elseif (t_1 <= -5e-93)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)) * J_m) * -2.0);
	elseif (t_1 <= -1e-291)
		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, -2e+304], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -4e+157], N[(N[(-0.25 * N[(U$95$m * N[(U$95$m / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * J$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-93], 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], If[LessEqual[t$95$1, -1e-291], 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 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))))) < -1.9999999999999999e304

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -1.9999999999999999e304 < (*.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))))) < -3.99999999999999993e157

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 29.2

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

   10. Applied rewrites 29.2%

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

   if -3.99999999999999993e157 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999994e-93

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   if -4.99999999999999994e-93 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999962e-292

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 29.5

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

   7. Applied rewrites 29.5%

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

   if -9.99999999999999962e-292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

       \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, U\_m \cdot \frac{U\_m}{J\_m \cdot J\_m}, -2\right) \cdot J\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(0.5, U\_m, \frac{J\_m \cdot J\_m}{U\_m}\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, -0.25 \cdot \frac{U\_m \cdot U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 -2e+304)
      (* (* U_m 0.5) -2.0)
      (if (<= t_1 -5e+35)
        (* (fma -0.25 (* U_m (/ U_m (* J_m J_m))) -2.0) J_m)
        (if (<= t_1 -5e-99)
          (* (fma 0.5 U_m (/ (* J_m J_m) U_m)) -2.0)
          (if (<= t_1 -1e-291)
            (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 <= -2e+304) {
		tmp = (U_m * 0.5) * -2.0;
	} else if (t_1 <= -5e+35) {
		tmp = fma(-0.25, (U_m * (U_m / (J_m * J_m))), -2.0) * J_m;
	} else if (t_1 <= -5e-99) {
		tmp = fma(0.5, U_m, ((J_m * J_m) / U_m)) * -2.0;
	} else if (t_1 <= -1e-291) {
		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 <= -2e+304)
		tmp = Float64(Float64(U_m * 0.5) * -2.0);
	elseif (t_1 <= -5e+35)
		tmp = Float64(fma(-0.25, Float64(U_m * Float64(U_m / Float64(J_m * J_m))), -2.0) * J_m);
	elseif (t_1 <= -5e-99)
		tmp = Float64(fma(0.5, U_m, Float64(Float64(J_m * J_m) / U_m)) * -2.0);
	elseif (t_1 <= -1e-291)
		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, -2e+304], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -5e+35], N[(N[(-0.25 * N[(U$95$m * N[(U$95$m / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision] * J$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-99], N[(N[(0.5 * U$95$m + N[(N[(J$95$m * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-291], 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 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))))) < -1.9999999999999999e304

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -1.9999999999999999e304 < (*.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))))) < -5.00000000000000021e35

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 29.2

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

   10. Applied rewrites 29.2%

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

   if -5.00000000000000021e35 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999969e-99

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   5. Taylor expanded in J around 0

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

      1. sqrt-prod N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 21.7

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

   7. Applied rewrites 21.7%

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

   8. Taylor expanded in U around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot {U}^{2} + {J}^{2}}{U} \cdot -2 \]

      2. *-commutative N/A

         \[\leadsto \frac{{U}^{2} \cdot \frac{1}{2} + {J}^{2}}{U} \cdot -2 \]

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 22.4

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

   10. Applied rewrites 22.4%

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

   11. Taylor expanded in J around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 37.1

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

   13. Applied rewrites 37.1%

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

   if -4.99999999999999969e-99 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999962e-292

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 29.5

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

   7. Applied rewrites 29.5%

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

   if -9.99999999999999962e-292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

       \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, -0.25 \cdot \frac{U\_m \cdot U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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 -2e+304)
      (* (* U_m 0.5) -2.0)
      (if (<= t_1 -1e-291)
        (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 <= -2e+304) {
		tmp = (U_m * 0.5) * -2.0;
	} else if (t_1 <= -1e-291) {
		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 <= -2e+304)
		tmp = Float64(Float64(U_m * 0.5) * -2.0);
	elseif (t_1 <= -1e-291)
		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, -2e+304], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-291], 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))))) < -1.9999999999999999e304

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -1.9999999999999999e304 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999962e-292

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 29.5

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

   7. Applied rewrites 29.5%

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

   if -9.99999999999999962e-292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

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

Alternative 10: 51.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}} \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\left(U\_m \cdot 0.5\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
         -1e-291)
      (* (* U_m 0.5) -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 tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -1e-291) {
		tmp = (U_m * 0.5) * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0)))) <= (-1d-291)) then
        tmp = (u_m * 0.5d0) * (-2.0d0)
    else
        tmp = ((-0.5d0) * u_m) * (-2.0d0)
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -1e-291) {
		tmp = (U_m * 0.5) * -2.0;
	} 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((K / 2.0))
	tmp = 0
	if (((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -1e-291:
		tmp = (U_m * 0.5) * -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))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0)))) <= -1e-291)
		tmp = Float64(Float64(U_m * 0.5) * -2.0);
	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((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)))) <= -1e-291)
		tmp = (U_m * 0.5) * -2.0;
	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[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-291], N[(N[(U$95$m * 0.5), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999962e-292

   1. Initial program 73.2%

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

   2. Taylor expanded in K around 0

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

      1. *-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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in U around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -9.99999999999999962e-292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.2%

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

   2. Taylor expanded in 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 34.1%

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

   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 21.9

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

   7. Applied rewrites 21.9%

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

   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 14.1

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

   10. Applied rewrites 14.1%

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

Alternative 11: 14.1% accurate, 15.9× speedup

Math

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

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 (* (* -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) {
	return J_s * ((-0.5 * U_m) * -2.0);
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * (((-0.5d0) * u_m) * (-2.0d0))
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * ((-0.5 * U_m) * -2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

2. Taylor expanded in 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 34.1%

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

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 21.9

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

7. Applied rewrites 21.9%

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

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 14.1

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

10. Applied rewrites 14.1%

    \[\leadsto \left(-0.5 \cdot U\right) \cdot -2 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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)))))