Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 99.5%

Time: 12.8s

Alternatives: 11

Speedup: 0.5×

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: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 -1e+308) (- U_m) (if (<= t_1 5e+302) t_1 U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double 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 <= -1e+308) {
		tmp = -U_m;
	} else if (t_1 <= 5e+302) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

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

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

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = (((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0)))
    if (t_1 <= (-1d+308)) then
        tmp = -u_m
    else if (t_1 <= 5d+302) then
        tmp = t_1
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -1e+308) {
		tmp = -U_m;
	} else if (t_1 <= 5e+302) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -1e+308:
		tmp = -U_m
	elif t_1 <= 5e+302:
		tmp = t_1
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	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 <= -1e+308)
		tmp = Float64(-U_m);
	elseif (t_1 <= 5e+302)
		tmp = t_1;
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -1e+308)
		tmp = -U_m;
	elseif (t_1 <= 5e+302)
		tmp = t_1;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := 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, -1e+308], (-U$95$m), If[LessEqual[t$95$1, 5e+302], t$95$1, U$95$m]]), $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))))) < -1e308

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

         \[\leadsto -U \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{-U} \]

   if -1e308 < (*.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))))) < 5e302

   1. Initial program 99.8%

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

   if 5e302 < (*.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 10.2%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Applied rewrites 45.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \cos \left(0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{-199}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{2 \cdot J\_m}\right)}^{2}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_3\right) \cdot \sqrt{\mathsf{fma}\left({\left(t\_3 \cdot J\_m\right)}^{-2} \cdot 0.25, U\_m \cdot U\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_3 (cos (* 0.5 K))))
   (*
    J_s
    (if (<= t_2 -1e+308)
      (- U_m)
      (if (<= t_2 1e-199)
        (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* 2.0 J_m)) 2.0))))
        (if (<= t_2 5e+302)
          (*
           (* (* -2.0 J_m) t_3)
           (sqrt (fma (* (pow (* t_3 J_m) -2.0) 0.25) (* U_m U_m) 1.0)))
          U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_3 = cos((0.5 * K));
	double tmp;
	if (t_2 <= -1e+308) {
		tmp = -U_m;
	} else if (t_2 <= 1e-199) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / (2.0 * J_m)), 2.0)));
	} else if (t_2 <= 5e+302) {
		tmp = ((-2.0 * J_m) * t_3) * sqrt(fma((pow((t_3 * J_m), -2.0) * 0.25), (U_m * U_m), 1.0));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_3 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_2 <= -1e+308)
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e-199)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(2.0 * J_m)) ^ 2.0))));
	elseif (t_2 <= 5e+302)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_3) * sqrt(fma(Float64((Float64(t_3 * J_m) ^ -2.0) * 0.25), Float64(U_m * U_m), 1.0)));
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -1e+308], (-U$95$m), If[LessEqual[t$95$2, 1e-199], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+302], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[N[(t$95$3 * J$95$m), $MachinePrecision], -2.0], $MachinePrecision] * 0.25), $MachinePrecision] * N[(U$95$m * U$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]), $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))))) < -1e308

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

         \[\leadsto -U \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{-U} \]

   if -1e308 < (*.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))))) < 9.99999999999999982e-200

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lift-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

   if 9.99999999999999982e-200 < (*.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))))) < 5e302

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \]
   4. 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 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in K around 0

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

      1. lift-*.f64 73.0

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

   8. Applied rewrites 73.0%

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

   9. Taylor expanded in U around inf

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

       1. distribute-rgt-in N/A

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

       2. inv-pow N/A

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

       3. pow-plus N/A

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

       4. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \sqrt{\left(\frac{1}{4} \cdot \frac{1}{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}\right) \cdot {U}^{2} + {\left({U}^{2}\right)}^{0}} \]

       5. metadata-eval N/A

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

       6. lower-fma.f64 N/A

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

   11. Applied rewrites 84.3%

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

   if 5e302 < (*.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 10.2%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Applied rewrites 45.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{-199}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{J\_m}{U\_m}, \frac{J\_m}{U\_m}, 0.5\right) \cdot \left(-U\_m\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -1e+308)
      (- U_m)
      (if (<= t_1 1e-199)
        (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
        (if (<= t_1 4e+58)
          (*
           (* (* -2.0 J_m) (cos (* 0.5 K)))
           (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)))
          (* (* (fma (/ J_m U_m) (/ J_m U_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 <= -1e+308) {
		tmp = -U_m;
	} else if (t_1 <= 1e-199) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_1 <= 4e+58) {
		tmp = ((-2.0 * J_m) * cos((0.5 * K))) * sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0));
	} else {
		tmp = (fma((J_m / U_m), (J_m / U_m), 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 <= -1e+308)
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e-199)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0);
	elseif (t_1 <= 4e+58)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K))) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)));
	else
		tmp = Float64(Float64(fma(Float64(J_m / U_m), Float64(J_m / U_m), 0.5) * Float64(-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, -1e+308], (-U$95$m), If[LessEqual[t$95$1, 1e-199], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 4e+58], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $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], N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * (-U$95$m)), $MachinePrecision] * -2.0), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e308

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

         \[\leadsto -U \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{-U} \]

   if -1e308 < (*.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))))) < 9.99999999999999982e-200

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 66.0%

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

   if 9.99999999999999982e-200 < (*.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))))) < 3.99999999999999978e58

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \]
   4. 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 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in K around 0

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

      1. lift-*.f64 72.7

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

   8. Applied rewrites 72.7%

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

   if 3.99999999999999978e58 < (*.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 63.2%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 40.3%

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

   8. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 26.1

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

   10. Applied rewrites 26.1%

       \[\leadsto \left(\left(-U\right) \cdot \left(\frac{J \cdot J}{U \cdot U} + 0.5\right)\right) \cdot -2 \]
   11. Step-by-step derivation

   12. Applied rewrites 26.6%

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

Alternative 4: 79.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-262}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{J\_m}{U\_m}, \frac{J\_m}{U\_m}, 0.5\right) \cdot \left(-U\_m\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -1e+308)
      (- U_m)
      (if (<= t_1 -5e-262)
        (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
        (if (<= t_1 4e+58)
          (* (* J_m -2.0) (cos (* 0.5 K)))
          (* (* (fma (/ J_m U_m) (/ J_m U_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 <= -1e+308) {
		tmp = -U_m;
	} else if (t_1 <= -5e-262) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_1 <= 4e+58) {
		tmp = (J_m * -2.0) * cos((0.5 * K));
	} else {
		tmp = (fma((J_m / U_m), (J_m / U_m), 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 <= -1e+308)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-262)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0);
	elseif (t_1 <= 4e+58)
		tmp = Float64(Float64(J_m * -2.0) * cos(Float64(0.5 * K)));
	else
		tmp = Float64(Float64(fma(Float64(J_m / U_m), Float64(J_m / U_m), 0.5) * Float64(-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, -1e+308], (-U$95$m), If[LessEqual[t$95$1, -5e-262], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 4e+58], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * (-U$95$m)), $MachinePrecision] * -2.0), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e308

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

         \[\leadsto -U \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{-U} \]

   if -1e308 < (*.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))))) < -4.99999999999999992e-262

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 66.5%

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

   if -4.99999999999999992e-262 < (*.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))))) < 3.99999999999999978e58

   1. Initial program 99.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   if 3.99999999999999978e58 < (*.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 63.2%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 40.3%

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

   8. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 26.1

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

   10. Applied rewrites 26.1%

       \[\leadsto \left(\left(-U\right) \cdot \left(\frac{J \cdot J}{U \cdot U} + 0.5\right)\right) \cdot -2 \]
   11. Step-by-step derivation

   12. Applied rewrites 26.6%

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

Alternative 5: 67.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+178}:\\ \;\;\;\;J\_m \cdot -2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-149}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{J\_m}{U\_m}, \frac{J\_m}{U\_m}, 0.5\right) \cdot \left(-U\_m\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -1e+308)
      (- U_m)
      (if (<= t_1 -1e+178)
        (* J_m -2.0)
        (if (<= t_1 -2e-149)
          (* (* J_m -2.0) (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)))
          (* (* (fma (/ J_m U_m) (/ J_m U_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 <= -1e+308) {
		tmp = -U_m;
	} else if (t_1 <= -1e+178) {
		tmp = J_m * -2.0;
	} else if (t_1 <= -2e-149) {
		tmp = (J_m * -2.0) * sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0));
	} else {
		tmp = (fma((J_m / U_m), (J_m / U_m), 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 <= -1e+308)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+178)
		tmp = Float64(J_m * -2.0);
	elseif (t_1 <= -2e-149)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)));
	else
		tmp = Float64(Float64(fma(Float64(J_m / U_m), Float64(J_m / U_m), 0.5) * Float64(-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, -1e+308], (-U$95$m), If[LessEqual[t$95$1, -1e+178], N[(J$95$m * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -2e-149], N[(N[(J$95$m * -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], N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * (-U$95$m)), $MachinePrecision] * -2.0), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e308

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

         \[\leadsto -U \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{-U} \]

   if -1e308 < (*.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))))) < -1.0000000000000001e178

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 35.8

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

   5. Applied rewrites 35.8%

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

   6. Taylor expanded in J around inf

      \[\leadsto -2 \cdot \color{blue}{J} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto J \cdot -2 \]

      2. lift-*.f64 40.3

         \[\leadsto J \cdot -2 \]

   8. Applied rewrites 40.3%

      \[\leadsto J \cdot \color{blue}{-2} \]

   if -1.0000000000000001e178 < (*.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))))) < -1.99999999999999996e-149

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   if -1.99999999999999996e-149 < (*.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 75.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.8

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

   5. Applied rewrites 34.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 47.8%

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

   8. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 22.1

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

   10. Applied rewrites 22.1%

       \[\leadsto \left(\left(-U\right) \cdot \left(\frac{J \cdot J}{U \cdot U} + 0.5\right)\right) \cdot -2 \]
   11. Step-by-step derivation

   12. Applied rewrites 24.0%

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

Alternative 6: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{2 \cdot J\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_2 -1e+308)
      (- U_m)
      (if (<= t_2 5e+302)
        (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* 2.0 J_m)) 2.0))))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -1e+308) {
		tmp = -U_m;
	} else if (t_2 <= 5e+302) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / (2.0 * J_m)), 2.0)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

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

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

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = ((-2.0d0) * j_m) * t_0
    t_2 = t_1 * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0)))
    if (t_2 <= (-1d+308)) then
        tmp = -u_m
    else if (t_2 <= 5d+302) then
        tmp = t_1 * sqrt((1.0d0 + ((u_m / (2.0d0 * j_m)) ** 2.0d0)))
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -1e+308) {
		tmp = -U_m;
	} else if (t_2 <= 5e+302) {
		tmp = t_1 * Math.sqrt((1.0 + Math.pow((U_m / (2.0 * J_m)), 2.0)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = (-2.0 * J_m) * t_0
	t_2 = t_1 * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	tmp = 0
	if t_2 <= -1e+308:
		tmp = -U_m
	elif t_2 <= 5e+302:
		tmp = t_1 * math.sqrt((1.0 + math.pow((U_m / (2.0 * J_m)), 2.0)))
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -1e+308)
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+302)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(2.0 * J_m)) ^ 2.0))));
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = (-2.0 * J_m) * t_0;
	t_2 = t_1 * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -1e+308)
		tmp = -U_m;
	elseif (t_2 <= 5e+302)
		tmp = t_1 * sqrt((1.0 + ((U_m / (2.0 * J_m)) ^ 2.0)));
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -1e+308], (-U$95$m), If[LessEqual[t$95$2, 5e+302], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]), $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))))) < -1e308

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

         \[\leadsto -U \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{-U} \]

   if -1e308 < (*.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))))) < 5e302

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lift-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if 5e302 < (*.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 10.2%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Applied rewrites 45.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-262}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{J\_m}{U\_m}, \frac{J\_m}{U\_m}, 0.5\right) \cdot \left(-U\_m\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -1e+308)
      (- U_m)
      (if (<= t_1 -5e-262)
        (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
        (* (* (fma (/ J_m U_m) (/ J_m U_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 <= -1e+308) {
		tmp = -U_m;
	} else if (t_1 <= -5e-262) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else {
		tmp = (fma((J_m / U_m), (J_m / U_m), 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 <= -1e+308)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-262)
		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(fma(Float64(J_m / U_m), Float64(J_m / U_m), 0.5) * Float64(-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, -1e+308], (-U$95$m), If[LessEqual[t$95$1, -5e-262], 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[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * (-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))))) < -1e308

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

         \[\leadsto -U \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{-U} \]

   if -1e308 < (*.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))))) < -4.99999999999999992e-262

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 66.5%

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

   if -4.99999999999999992e-262 < (*.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 74.6%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 36.6

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

   5. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 46.5%

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

   8. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 23.3

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

   10. Applied rewrites 23.3%

       \[\leadsto \left(\left(-U\right) \cdot \left(\frac{J \cdot J}{U \cdot U} + 0.5\right)\right) \cdot -2 \]
   11. Step-by-step derivation

   12. Applied rewrites 23.7%

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

Alternative 8: 62.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-262}:\\ \;\;\;\;J\_m \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{J\_m}{U\_m}, \frac{J\_m}{U\_m}, 0.5\right) \cdot \left(-U\_m\right)\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -1e+308)
      (- U_m)
      (if (<= t_1 -5e-262)
        (* J_m -2.0)
        (* (* (fma (/ J_m U_m) (/ J_m U_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 <= -1e+308) {
		tmp = -U_m;
	} else if (t_1 <= -5e-262) {
		tmp = J_m * -2.0;
	} else {
		tmp = (fma((J_m / U_m), (J_m / U_m), 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 <= -1e+308)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-262)
		tmp = Float64(J_m * -2.0);
	else
		tmp = Float64(Float64(fma(Float64(J_m / U_m), Float64(J_m / U_m), 0.5) * Float64(-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, -1e+308], (-U$95$m), If[LessEqual[t$95$1, -5e-262], N[(J$95$m * -2.0), $MachinePrecision], N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * (-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))))) < -1e308

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

         \[\leadsto -U \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{-U} \]

   if -1e308 < (*.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))))) < -4.99999999999999992e-262

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in J around inf

      \[\leadsto -2 \cdot \color{blue}{J} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto J \cdot -2 \]

      2. lift-*.f64 40.8

         \[\leadsto J \cdot -2 \]

   8. Applied rewrites 40.8%

      \[\leadsto J \cdot \color{blue}{-2} \]

   if -4.99999999999999992e-262 < (*.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 74.6%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 36.6

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

   5. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

   7. Applied rewrites 46.5%

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

   8. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 23.3

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

   10. Applied rewrites 23.3%

       \[\leadsto \left(\left(-U\right) \cdot \left(\frac{J \cdot J}{U \cdot U} + 0.5\right)\right) \cdot -2 \]
   11. Step-by-step derivation

   12. Applied rewrites 23.7%

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

Alternative 9: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-262}:\\ \;\;\;\;J\_m \cdot -2\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 -1e+308) (- U_m) (if (<= t_1 -5e-262) (* J_m -2.0) U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double 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 <= -1e+308) {
		tmp = -U_m;
	} else if (t_1 <= -5e-262) {
		tmp = J_m * -2.0;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Fortran

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

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

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = (((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0)))
    if (t_1 <= (-1d+308)) then
        tmp = -u_m
    else if (t_1 <= (-5d-262)) then
        tmp = j_m * (-2.0d0)
    else
        tmp = u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -1e+308) {
		tmp = -U_m;
	} else if (t_1 <= -5e-262) {
		tmp = J_m * -2.0;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -1e+308:
		tmp = -U_m
	elif t_1 <= -5e-262:
		tmp = J_m * -2.0
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	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 <= -1e+308)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-262)
		tmp = Float64(J_m * -2.0);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -1e+308)
		tmp = -U_m;
	elseif (t_1 <= -5e-262)
		tmp = J_m * -2.0;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := 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, -1e+308], (-U$95$m), If[LessEqual[t$95$1, -5e-262], N[(J$95$m * -2.0), $MachinePrecision], U$95$m]]), $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))))) < -1e308

   1. Initial program 6.1%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

         \[\leadsto -U \]

   5. Applied rewrites 39.3%

      \[\leadsto \color{blue}{-U} \]

   if -1e308 < (*.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))))) < -4.99999999999999992e-262

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in J around inf

      \[\leadsto -2 \cdot \color{blue}{J} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto J \cdot -2 \]

      2. lift-*.f64 40.8

         \[\leadsto J \cdot -2 \]

   8. Applied rewrites 40.8%

      \[\leadsto J \cdot \color{blue}{-2} \]

   if -4.99999999999999992e-262 < (*.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 74.6%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Applied rewrites 23.6%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\_m\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= (cos (/ K 2.0)) -4e-310) U_m (- U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (cos((K / 2.0)) <= -4e-310) {
		tmp = U_m;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Fortran

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

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

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-4d-310)) then
        tmp = u_m
    else
        tmp = -u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -4e-310) {
		tmp = U_m;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if math.cos((K / 2.0)) <= -4e-310:
		tmp = U_m
	else:
		tmp = -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -4e-310)
		tmp = U_m;
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -4e-310)
		tmp = U_m;
	else
		tmp = -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -4e-310], U$95$m, (-U$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -3.999999999999988e-310

   1. Initial program 72.3%

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

   3. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Applied rewrites 39.4%

      \[\leadsto \color{blue}{U} \]

   if -3.999999999999988e-310 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 73.7%

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

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.2

         \[\leadsto -U \]

   5. Applied rewrites 26.2%

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

Alternative 11: 13.6% accurate, 373.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in U around -inf

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Applied rewrites 29.5%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025043 
(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)))))