Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.8% → 99.0%

Time: 6.5s

Alternatives: 13

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: 73.8% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+293}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot \cos \left(0.5 \cdot K\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (- INFINITY))
      (- U_m)
      (if (<= t_2 1e+293)
        (*
         t_1
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) (cos (* 0.5 K)))) 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 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+293) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * cos((0.5 * K)))), 2.0)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double 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 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 1e+293) {
		tmp = t_1 * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * Math.cos((0.5 * K)))), 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 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 1e+293:
		tmp = t_1 * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * math.cos((0.5 * K)))), 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 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+293)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * cos(Float64(0.5 * K)))) ^ 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 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 1e+293)
		tmp = t_1 * sqrt((1.0 + ((U_m / ((2.0 * J_m) * cos((0.5 * K)))) ^ 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, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+293], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 5.6%

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

   3. Taylor expanded in 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 64.3

         \[\leadsto -U \]

   5. Applied rewrites 64.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999992e292

   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

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

      1. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 9.9999999999999992e292 < (*.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 13.3%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 45.5%

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

Alternative 2: 85.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \cos \left(0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;U\_m \cdot \left(-2 \cdot \left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right) - 1\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.25, t\_3 \cdot -2\right) \cdot J\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_3\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\left(-J\_m\right) \cdot \left(\sqrt{1 - -0.25 \cdot t\_0} + \mathsf{fma}\left(0.5, \frac{U\_m}{J\_m}, \frac{1}{\sqrt{1 - -0.25 \cdot {\left(\frac{U\_m}{J\_m}\right)}^{2}} - -0.5 \cdot \frac{U\_m}{J\_m}}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+293}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (/ U_m J_m) (/ U_m J_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0)))))
        (t_3 (cos (* 0.5 K))))
   (*
    J_s
    (if (<= t_2 -4e+300)
      (* U_m (- (* -2.0 (* (/ J_m U_m) (/ J_m U_m))) 1.0))
      (if (<= t_2 -2e+168)
        (* (fma t_0 -0.25 (* t_3 -2.0)) J_m)
        (if (<= t_2 -2e-30)
          (*
           (* (* -2.0 J_m) t_3)
           (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)))
          (if (<= t_2 -5e-271)
            (*
             (- J_m)
             (+
              (sqrt (- 1.0 (* -0.25 t_0)))
              (fma
               0.5
               (/ U_m J_m)
               (/
                1.0
                (-
                 (sqrt (- 1.0 (* -0.25 (pow (/ U_m J_m) 2.0))))
                 (* -0.5 (/ U_m J_m)))))))
            (if (<= t_2 1e+293) (* (* J_m -2.0) t_3) 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 = (U_m / J_m) * (U_m / J_m);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double t_3 = cos((0.5 * K));
	double tmp;
	if (t_2 <= -4e+300) {
		tmp = U_m * ((-2.0 * ((J_m / U_m) * (J_m / U_m))) - 1.0);
	} else if (t_2 <= -2e+168) {
		tmp = fma(t_0, -0.25, (t_3 * -2.0)) * J_m;
	} else if (t_2 <= -2e-30) {
		tmp = ((-2.0 * J_m) * t_3) * sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0));
	} else if (t_2 <= -5e-271) {
		tmp = -J_m * (sqrt((1.0 - (-0.25 * t_0))) + fma(0.5, (U_m / J_m), (1.0 / (sqrt((1.0 - (-0.25 * pow((U_m / J_m), 2.0)))) - (-0.5 * (U_m / J_m))))));
	} else if (t_2 <= 1e+293) {
		tmp = (J_m * -2.0) * t_3;
	} 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 = Float64(Float64(U_m / J_m) * Float64(U_m / J_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	t_3 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_2 <= -4e+300)
		tmp = Float64(U_m * Float64(Float64(-2.0 * Float64(Float64(J_m / U_m) * Float64(J_m / U_m))) - 1.0));
	elseif (t_2 <= -2e+168)
		tmp = Float64(fma(t_0, -0.25, Float64(t_3 * -2.0)) * J_m);
	elseif (t_2 <= -2e-30)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_3) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)));
	elseif (t_2 <= -5e-271)
		tmp = Float64(Float64(-J_m) * Float64(sqrt(Float64(1.0 - Float64(-0.25 * t_0))) + fma(0.5, Float64(U_m / J_m), Float64(1.0 / Float64(sqrt(Float64(1.0 - Float64(-0.25 * (Float64(U_m / J_m) ^ 2.0)))) - Float64(-0.5 * Float64(U_m / J_m)))))));
	elseif (t_2 <= 1e+293)
		tmp = Float64(Float64(J_m * -2.0) * t_3);
	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[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -4e+300], N[(U$95$m * N[(N[(-2.0 * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+168], N[(N[(t$95$0 * -0.25 + N[(t$95$3 * -2.0), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision], If[LessEqual[t$95$2, -2e-30], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-271], N[((-J$95$m) * N[(N[Sqrt[N[(1.0 - N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(1.0 - N[(-0.25 * N[Power[N[(U$95$m / J$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(-0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+293], N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$3), $MachinePrecision], U$95$m]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 6 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))))) < -4.0000000000000002e300

   1. Initial program 10.1%

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

   3. Taylor expanded in 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 4.2

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 34.6

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in U around inf

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

       1. lower--.f64 N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 61.2

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. times-frac N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 63.6

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

   11. Applied rewrites 63.6%

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

   if -4.0000000000000002e300 < (*.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))))) < -1.9999999999999999e168

   1. Initial program 99.9%

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

   3. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in K around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.5

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

   8. Applied rewrites 87.5%

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

   if -1.9999999999999999e168 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2e-30

   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 93.1

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

   5. Applied rewrites 93.1%

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

   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 93.1

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

   8. Applied rewrites 93.1%

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

   if -2e-30 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-271

   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. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. cosh-asinh-rev N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-asinh.f64 N/A

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

   7. Applied rewrites 40.0%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. unpow-1 N/A

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

      12. lower-/.f64 N/A

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

   9. Applied rewrites 40.0%

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

   if -5.0000000000000002e-271 < (*.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))))) < 9.9999999999999992e292

   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 77.7

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

   5. Applied rewrites 77.7%

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

   if 9.9999999999999992e292 < (*.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 13.3%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 45.5%

      \[\leadsto \color{blue}{U} \]
3. Recombined 6 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -4 \cdot 10^{+300}:\\ \;\;\;\;U \cdot \left(-2 \cdot \left(\frac{J}{U} \cdot \frac{J}{U}\right) - 1\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, -0.25, \cos \left(0.5 \cdot K\right) \cdot -2\right) \cdot J\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -2 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\left(-J\right) \cdot \left(\sqrt{1 - -0.25 \cdot \left(\frac{U}{J} \cdot \frac{U}{J}\right)} + \mathsf{fma}\left(0.5, \frac{U}{J}, \frac{1}{\sqrt{1 - -0.25 \cdot {\left(\frac{U}{J}\right)}^{2}} - -0.5 \cdot \frac{U}{J}}\right)\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq 10^{+293}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \cos \left(0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;U\_m \cdot \left(-2 \cdot \left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right) - 1\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.25, t\_3 \cdot -2\right) \cdot J\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(-2 \cdot J\_m\right) \cdot t\_3\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+293}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (/ U_m J_m) (/ U_m J_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0)))))
        (t_3 (cos (* 0.5 K))))
   (*
    J_s
    (if (<= t_2 -4e+300)
      (* U_m (- (* -2.0 (* (/ J_m U_m) (/ J_m U_m))) 1.0))
      (if (<= t_2 -2e+168)
        (* (fma t_0 -0.25 (* t_3 -2.0)) J_m)
        (if (<= t_2 -2e-30)
          (*
           (* (* -2.0 J_m) t_3)
           (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)))
          (if (<= t_2 -5e-271)
            (* (* J_m -2.0) (sqrt (fma t_0 0.25 1.0)))
            (if (<= t_2 1e+293) (* (* J_m -2.0) t_3) 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 = (U_m / J_m) * (U_m / J_m);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double t_3 = cos((0.5 * K));
	double tmp;
	if (t_2 <= -4e+300) {
		tmp = U_m * ((-2.0 * ((J_m / U_m) * (J_m / U_m))) - 1.0);
	} else if (t_2 <= -2e+168) {
		tmp = fma(t_0, -0.25, (t_3 * -2.0)) * J_m;
	} else if (t_2 <= -2e-30) {
		tmp = ((-2.0 * J_m) * t_3) * sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0));
	} else if (t_2 <= -5e-271) {
		tmp = (J_m * -2.0) * sqrt(fma(t_0, 0.25, 1.0));
	} else if (t_2 <= 1e+293) {
		tmp = (J_m * -2.0) * t_3;
	} 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 = Float64(Float64(U_m / J_m) * Float64(U_m / J_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	t_3 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_2 <= -4e+300)
		tmp = Float64(U_m * Float64(Float64(-2.0 * Float64(Float64(J_m / U_m) * Float64(J_m / U_m))) - 1.0));
	elseif (t_2 <= -2e+168)
		tmp = Float64(fma(t_0, -0.25, Float64(t_3 * -2.0)) * J_m);
	elseif (t_2 <= -2e-30)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_3) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)));
	elseif (t_2 <= -5e-271)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(t_0, 0.25, 1.0)));
	elseif (t_2 <= 1e+293)
		tmp = Float64(Float64(J_m * -2.0) * t_3);
	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[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -4e+300], N[(U$95$m * N[(N[(-2.0 * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e+168], N[(N[(t$95$0 * -0.25 + N[(t$95$3 * -2.0), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision], If[LessEqual[t$95$2, -2e-30], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-271], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(t$95$0 * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+293], N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$3), $MachinePrecision], U$95$m]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 6 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))))) < -4.0000000000000002e300

   1. Initial program 10.1%

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

   3. Taylor expanded in 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 4.2

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 34.6

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in U around inf

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

       1. lower--.f64 N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 61.2

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. times-frac N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 63.6

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

   11. Applied rewrites 63.6%

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

   if -4.0000000000000002e300 < (*.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))))) < -1.9999999999999999e168

   1. Initial program 99.9%

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

   3. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in K around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.5

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

   8. Applied rewrites 87.5%

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

   if -1.9999999999999999e168 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2e-30

   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 93.1

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

   5. Applied rewrites 93.1%

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

   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 93.1

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

   8. Applied rewrites 93.1%

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

   if -2e-30 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-271

   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 32.2

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

   5. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 65.7

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

   7. Applied rewrites 65.7%

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

   if -5.0000000000000002e-271 < (*.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))))) < 9.9999999999999992e292

   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 77.7

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

   5. Applied rewrites 77.7%

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

   if 9.9999999999999992e292 < (*.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 13.3%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 45.5%

      \[\leadsto \color{blue}{U} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \cos \left(0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;U\_m \cdot \left(-2 \cdot \left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right) - 1\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.25, t\_3 \cdot -2\right) \cdot J\_m\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+293}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (/ U_m J_m) (/ U_m J_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0)))))
        (t_3 (cos (* 0.5 K))))
   (*
    J_s
    (if (<= t_2 -4e+300)
      (* U_m (- (* -2.0 (* (/ J_m U_m) (/ J_m U_m))) 1.0))
      (if (<= t_2 -4e+34)
        (* (fma t_0 -0.25 (* t_3 -2.0)) J_m)
        (if (<= t_2 -5e-271)
          (* (* J_m -2.0) (sqrt (fma t_0 0.25 1.0)))
          (if (<= t_2 1e+293) (* (* J_m -2.0) t_3) 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 = (U_m / J_m) * (U_m / J_m);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double t_3 = cos((0.5 * K));
	double tmp;
	if (t_2 <= -4e+300) {
		tmp = U_m * ((-2.0 * ((J_m / U_m) * (J_m / U_m))) - 1.0);
	} else if (t_2 <= -4e+34) {
		tmp = fma(t_0, -0.25, (t_3 * -2.0)) * J_m;
	} else if (t_2 <= -5e-271) {
		tmp = (J_m * -2.0) * sqrt(fma(t_0, 0.25, 1.0));
	} else if (t_2 <= 1e+293) {
		tmp = (J_m * -2.0) * t_3;
	} 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 = Float64(Float64(U_m / J_m) * Float64(U_m / J_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	t_3 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_2 <= -4e+300)
		tmp = Float64(U_m * Float64(Float64(-2.0 * Float64(Float64(J_m / U_m) * Float64(J_m / U_m))) - 1.0));
	elseif (t_2 <= -4e+34)
		tmp = Float64(fma(t_0, -0.25, Float64(t_3 * -2.0)) * J_m);
	elseif (t_2 <= -5e-271)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(t_0, 0.25, 1.0)));
	elseif (t_2 <= 1e+293)
		tmp = Float64(Float64(J_m * -2.0) * t_3);
	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[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -4e+300], N[(U$95$m * N[(N[(-2.0 * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e+34], N[(N[(t$95$0 * -0.25 + N[(t$95$3 * -2.0), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision], If[LessEqual[t$95$2, -5e-271], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(t$95$0 * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+293], N[(N[(J$95$m * -2.0), $MachinePrecision] * t$95$3), $MachinePrecision], U$95$m]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000002e300

   1. Initial program 10.1%

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

   3. Taylor expanded in 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 4.2

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 34.6

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in U around inf

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

       1. lower--.f64 N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 61.2

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. times-frac N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 63.6

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

   11. Applied rewrites 63.6%

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

   if -4.0000000000000002e300 < (*.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))))) < -3.99999999999999978e34

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in K around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 82.9

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

   8. Applied rewrites 82.9%

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

   if -3.99999999999999978e34 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-271

   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 44.9

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

   5. Applied rewrites 44.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 66.1

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

   7. Applied rewrites 66.1%

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

   if -5.0000000000000002e-271 < (*.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))))) < 9.9999999999999992e292

   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 77.7

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

   5. Applied rewrites 77.7%

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

   if 9.9999999999999992e292 < (*.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 13.3%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 45.5%

      \[\leadsto \color{blue}{U} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(J\_m \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;U\_m \cdot \left(-2 \cdot \left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right) - 1\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+293}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* J_m -2.0) (cos (* 0.5 K))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 -4e+300)
      (* U_m (- (* -2.0 (* (/ J_m U_m) (/ J_m U_m))) 1.0))
      (if (<= t_2 -4e+34)
        t_0
        (if (<= t_2 -5e-271)
          (* (* J_m -2.0) (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)))
          (if (<= t_2 1e+293) t_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 = (J_m * -2.0) * cos((0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -4e+300) {
		tmp = U_m * ((-2.0 * ((J_m / U_m) * (J_m / U_m))) - 1.0);
	} else if (t_2 <= -4e+34) {
		tmp = t_0;
	} else if (t_2 <= -5e-271) {
		tmp = (J_m * -2.0) * sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0));
	} else if (t_2 <= 1e+293) {
		tmp = t_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 = Float64(Float64(J_m * -2.0) * cos(Float64(0.5 * K)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -4e+300)
		tmp = Float64(U_m * Float64(Float64(-2.0 * Float64(Float64(J_m / U_m) * Float64(J_m / U_m))) - 1.0));
	elseif (t_2 <= -4e+34)
		tmp = t_0;
	elseif (t_2 <= -5e-271)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)));
	elseif (t_2 <= 1e+293)
		tmp = t_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[(N[(J$95$m * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -4e+300], N[(U$95$m * N[(N[(-2.0 * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e+34], t$95$0, If[LessEqual[t$95$2, -5e-271], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+293], t$95$0, 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))))) < -4.0000000000000002e300

   1. Initial program 10.1%

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

   3. Taylor expanded in 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 4.2

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 34.6

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in U around inf

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

       1. lower--.f64 N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 61.2

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. times-frac N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 63.6

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

   11. Applied rewrites 63.6%

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

   if -4.0000000000000002e300 < (*.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))))) < -3.99999999999999978e34 or -5.0000000000000002e-271 < (*.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))))) < 9.9999999999999992e292

   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 80.0

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

   5. Applied rewrites 80.0%

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

   if -3.99999999999999978e34 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-271

   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 44.9

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

   5. Applied rewrites 44.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 66.1

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

   7. Applied rewrites 66.1%

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

   if 9.9999999999999992e292 < (*.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 13.3%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 45.5%

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

Alternative 6: 66.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := -2 \cdot \left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right)\\ t_1 := U\_m \cdot \left(t\_0 - 1\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+278}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (/ J_m U_m) (/ J_m U_m))))
        (t_1 (* U_m (- t_0 1.0)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0))))))
   (*
    J_s
    (if (<= t_3 -1e+278)
      t_1
      (if (<= t_3 -1e-90)
        (* (* J_m -2.0) (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)))
        (if (<= t_3 -5e-271) t_1 (* U_m (- 1.0 t_0))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = -2.0 * ((J_m / U_m) * (J_m / U_m));
	double t_1 = U_m * (t_0 - 1.0);
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -1e+278) {
		tmp = t_1;
	} else if (t_3 <= -1e-90) {
		tmp = (J_m * -2.0) * sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0));
	} else if (t_3 <= -5e-271) {
		tmp = t_1;
	} else {
		tmp = U_m * (1.0 - t_0);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(-2.0 * Float64(Float64(J_m / U_m) * Float64(J_m / U_m)))
	t_1 = Float64(U_m * Float64(t_0 - 1.0))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -1e+278)
		tmp = t_1;
	elseif (t_3 <= -1e-90)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)));
	elseif (t_3 <= -5e-271)
		tmp = t_1;
	else
		tmp = Float64(U_m * Float64(1.0 - t_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[(-2.0 * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U$95$m * N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, -1e+278], t$95$1, If[LessEqual[t$95$3, -1e-90], 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], If[LessEqual[t$95$3, -5e-271], t$95$1, N[(U$95$m * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999964e277 or -9.99999999999999995e-91 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-271

   1. Initial program 31.7%

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

   3. Taylor expanded in 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 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 26.4

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

   8. Applied rewrites 26.4%

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

   9. Taylor expanded in U around inf

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

       1. lower--.f64 N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 50.3

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. times-frac N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 53.7

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

   11. Applied rewrites 53.7%

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

   if -9.99999999999999964e277 < (*.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))))) < -9.99999999999999995e-91

   1. Initial program 99.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   if -5.0000000000000002e-271 < (*.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 65.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 26.2

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

   5. Applied rewrites 26.2%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 24.7

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

   8. Applied rewrites 24.7%

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

   9. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. lower--.f64 N/A

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

       5. pow2 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 N/A

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

       8. times-frac N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 28.6

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

   11. Applied rewrites 28.6%

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

Alternative 7: 62.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := -2 \cdot \left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right)\\ t_1 := U\_m \cdot \left(t\_0 - 1\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_2}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-90}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m \cdot \left(1 - t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (/ J_m U_m) (/ J_m U_m))))
        (t_1 (* U_m (- t_0 1.0)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_2)) 2.0))))))
   (*
    J_s
    (if (<= t_3 -4e+300)
      t_1
      (if (<= t_3 -1e-90)
        (* -2.0 J_m)
        (if (<= t_3 -5e-271) t_1 (* U_m (- 1.0 t_0))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = -2.0 * ((J_m / U_m) * (J_m / U_m));
	double t_1 = U_m * (t_0 - 1.0);
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -4e+300) {
		tmp = t_1;
	} else if (t_3 <= -1e-90) {
		tmp = -2.0 * J_m;
	} else if (t_3 <= -5e-271) {
		tmp = t_1;
	} else {
		tmp = U_m * (1.0 - t_0);
	}
	return J_s * tmp;
}

Fortran

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

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

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (-2.0d0) * ((j_m / u_m) * (j_m / u_m))
    t_1 = u_m * (t_0 - 1.0d0)
    t_2 = cos((k / 2.0d0))
    t_3 = (((-2.0d0) * j_m) * t_2) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_2)) ** 2.0d0)))
    if (t_3 <= (-4d+300)) then
        tmp = t_1
    else if (t_3 <= (-1d-90)) then
        tmp = (-2.0d0) * j_m
    else if (t_3 <= (-5d-271)) then
        tmp = t_1
    else
        tmp = u_m * (1.0d0 - t_0)
    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 = -2.0 * ((J_m / U_m) * (J_m / U_m));
	double t_1 = U_m * (t_0 - 1.0);
	double t_2 = Math.cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -4e+300) {
		tmp = t_1;
	} else if (t_3 <= -1e-90) {
		tmp = -2.0 * J_m;
	} else if (t_3 <= -5e-271) {
		tmp = t_1;
	} else {
		tmp = U_m * (1.0 - t_0);
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = -2.0 * ((J_m / U_m) * (J_m / U_m))
	t_1 = U_m * (t_0 - 1.0)
	t_2 = math.cos((K / 2.0))
	t_3 = ((-2.0 * J_m) * t_2) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_2)), 2.0)))
	tmp = 0
	if t_3 <= -4e+300:
		tmp = t_1
	elif t_3 <= -1e-90:
		tmp = -2.0 * J_m
	elif t_3 <= -5e-271:
		tmp = t_1
	else:
		tmp = U_m * (1.0 - t_0)
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(-2.0 * Float64(Float64(J_m / U_m) * Float64(J_m / U_m)))
	t_1 = Float64(U_m * Float64(t_0 - 1.0))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -4e+300)
		tmp = t_1;
	elseif (t_3 <= -1e-90)
		tmp = Float64(-2.0 * J_m);
	elseif (t_3 <= -5e-271)
		tmp = t_1;
	else
		tmp = Float64(U_m * Float64(1.0 - t_0));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = -2.0 * ((J_m / U_m) * (J_m / U_m));
	t_1 = U_m * (t_0 - 1.0);
	t_2 = cos((K / 2.0));
	t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_2)) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -4e+300)
		tmp = t_1;
	elseif (t_3 <= -1e-90)
		tmp = -2.0 * J_m;
	elseif (t_3 <= -5e-271)
		tmp = t_1;
	else
		tmp = U_m * (1.0 - t_0);
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(-2.0 * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U$95$m * N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, -4e+300], t$95$1, If[LessEqual[t$95$3, -1e-90], N[(-2.0 * J$95$m), $MachinePrecision], If[LessEqual[t$95$3, -5e-271], t$95$1, N[(U$95$m * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000002e300 or -9.99999999999999995e-91 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-271

   1. Initial program 26.3%

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

   3. Taylor expanded in 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 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 28.5

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

   8. Applied rewrites 28.5%

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

   9. Taylor expanded in U around inf

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

       1. lower--.f64 N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 54.2

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. times-frac N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 57.4

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

   11. Applied rewrites 57.4%

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

   if -4.0000000000000002e300 < (*.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))))) < -9.99999999999999995e-91

   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 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in J around inf

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

      1. lift-*.f64 39.8

         \[\leadsto -2 \cdot J \]

   8. Applied rewrites 39.8%

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

   if -5.0000000000000002e-271 < (*.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 65.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 26.2

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

   5. Applied rewrites 26.2%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 24.7

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

   8. Applied rewrites 24.7%

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

   9. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. lower--.f64 N/A

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

       5. pow2 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 N/A

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

       8. times-frac N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 28.6

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

   11. Applied rewrites 28.6%

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

Alternative 8: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-90}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m \cdot \left(1 - -2 \cdot \left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (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 -4e+300)
      (- U_m)
      (if (<= t_1 -1e-90)
        (* -2.0 J_m)
        (if (<= t_1 -5e-271)
          (- U_m)
          (* U_m (- 1.0 (* -2.0 (* (/ J_m U_m) (/ J_m U_m)))))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double 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 <= -4e+300) {
		tmp = -U_m;
	} else if (t_1 <= -1e-90) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -5e-271) {
		tmp = -U_m;
	} else {
		tmp = U_m * (1.0 - (-2.0 * ((J_m / U_m) * (J_m / 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 <= (-4d+300)) then
        tmp = -u_m
    else if (t_1 <= (-1d-90)) then
        tmp = (-2.0d0) * j_m
    else if (t_1 <= (-5d-271)) then
        tmp = -u_m
    else
        tmp = u_m * (1.0d0 - ((-2.0d0) * ((j_m / u_m) * (j_m / 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 <= -4e+300) {
		tmp = -U_m;
	} else if (t_1 <= -1e-90) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -5e-271) {
		tmp = -U_m;
	} else {
		tmp = U_m * (1.0 - (-2.0 * ((J_m / U_m) * (J_m / 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 <= -4e+300:
		tmp = -U_m
	elif t_1 <= -1e-90:
		tmp = -2.0 * J_m
	elif t_1 <= -5e-271:
		tmp = -U_m
	else:
		tmp = U_m * (1.0 - (-2.0 * ((J_m / U_m) * (J_m / U_m))))
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	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 <= -4e+300)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-90)
		tmp = Float64(-2.0 * J_m);
	elseif (t_1 <= -5e-271)
		tmp = Float64(-U_m);
	else
		tmp = Float64(U_m * Float64(1.0 - Float64(-2.0 * Float64(Float64(J_m / U_m) * Float64(J_m / 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 <= -4e+300)
		tmp = -U_m;
	elseif (t_1 <= -1e-90)
		tmp = -2.0 * J_m;
	elseif (t_1 <= -5e-271)
		tmp = -U_m;
	else
		tmp = U_m * (1.0 - (-2.0 * ((J_m / U_m) * (J_m / 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, -4e+300], (-U$95$m), If[LessEqual[t$95$1, -1e-90], N[(-2.0 * J$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-271], (-U$95$m), N[(U$95$m * N[(1.0 - N[(-2.0 * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000002e300 or -9.99999999999999995e-91 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-271

   1. Initial program 26.3%

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

   3. Taylor expanded in 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 57.3

         \[\leadsto -U \]

   5. Applied rewrites 57.3%

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

   if -4.0000000000000002e300 < (*.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))))) < -9.99999999999999995e-91

   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 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in J around inf

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

      1. lift-*.f64 39.8

         \[\leadsto -2 \cdot J \]

   8. Applied rewrites 39.8%

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

   if -5.0000000000000002e-271 < (*.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 65.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 26.2

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

   5. Applied rewrites 26.2%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 24.7

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

   8. Applied rewrites 24.7%

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

   9. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. lower--.f64 N/A

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

       5. pow2 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 N/A

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

       8. times-frac N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 28.6

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

   11. Applied rewrites 28.6%

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

Alternative 9: 62.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-90}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 -4e+300)
      (- U_m)
      (if (<= t_1 -1e-90) (* -2.0 J_m) (if (<= t_1 -5e-271) (- 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 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 <= -4e+300) {
		tmp = -U_m;
	} else if (t_1 <= -1e-90) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -5e-271) {
		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) :: 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 <= (-4d+300)) then
        tmp = -u_m
    else if (t_1 <= (-1d-90)) then
        tmp = (-2.0d0) * j_m
    else if (t_1 <= (-5d-271)) 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 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 <= -4e+300) {
		tmp = -U_m;
	} else if (t_1 <= -1e-90) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -5e-271) {
		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):
	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 <= -4e+300:
		tmp = -U_m
	elif t_1 <= -1e-90:
		tmp = -2.0 * J_m
	elif t_1 <= -5e-271:
		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)
	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 <= -4e+300)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-90)
		tmp = Float64(-2.0 * J_m);
	elseif (t_1 <= -5e-271)
		tmp = Float64(-U_m);
	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 <= -4e+300)
		tmp = -U_m;
	elseif (t_1 <= -1e-90)
		tmp = -2.0 * J_m;
	elseif (t_1 <= -5e-271)
		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_] := 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, -4e+300], (-U$95$m), If[LessEqual[t$95$1, -1e-90], N[(-2.0 * J$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-271], (-U$95$m), 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))))) < -4.0000000000000002e300 or -9.99999999999999995e-91 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-271

   1. Initial program 26.3%

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

   3. Taylor expanded in 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 57.3

         \[\leadsto -U \]

   5. Applied rewrites 57.3%

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

   if -4.0000000000000002e300 < (*.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))))) < -9.99999999999999995e-91

   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 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in J around inf

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

      1. lift-*.f64 39.8

         \[\leadsto -2 \cdot J \]

   8. Applied rewrites 39.8%

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

   if -5.0000000000000002e-271 < (*.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 65.6%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 28.4%

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

Alternative 10: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+293}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{2 \cdot J\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

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 (- INFINITY))
      (- U_m)
      (if (<= t_2 1e+293)
        (* 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 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+293) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / (2.0 * J_m)), 2.0)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double 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 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 1e+293) {
		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 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 1e+293:
		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 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+293)
		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 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 1e+293)
		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, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+293], 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))))) < -inf.0

   1. Initial program 5.6%

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

   3. Taylor expanded in 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 64.3

         \[\leadsto -U \]

   5. Applied rewrites 64.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999992e292

   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 88.3

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

   5. Applied rewrites 88.3%

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

   if 9.9999999999999992e292 < (*.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 13.3%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 45.5%

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

Alternative 11: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m \cdot \left(1 - -2 \cdot \left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -5e-271)
        (* (* J_m -2.0) (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)))
        (* U_m (- 1.0 (* -2.0 (* (/ J_m U_m) (/ J_m U_m))))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e-271) {
		tmp = (J_m * -2.0) * sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0));
	} else {
		tmp = U_m * (1.0 - (-2.0 * ((J_m / U_m) * (J_m / U_m))));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-271)
		tmp = Float64(Float64(J_m * -2.0) * sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)));
	else
		tmp = Float64(U_m * Float64(1.0 - Float64(-2.0 * Float64(Float64(J_m / U_m) * Float64(J_m / U_m)))));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -5e-271], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(U$95$m * N[(1.0 - N[(-2.0 * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in 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 64.3

         \[\leadsto -U \]

   5. Applied rewrites 64.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000002e-271

   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 49.6

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

   5. Applied rewrites 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 62.0

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

   7. Applied rewrites 62.0%

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

   if -5.0000000000000002e-271 < (*.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 65.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 26.2

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

   5. Applied rewrites 26.2%

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

   6. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 24.7

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

   8. Applied rewrites 24.7%

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

   9. Taylor expanded in U around inf

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

       1. lower-*.f64 N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. lower--.f64 N/A

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

       5. pow2 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 N/A

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

       8. times-frac N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 28.6

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

   11. Applied rewrites 28.6%

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

Alternative 12: 51.0% accurate, 3.1× speedup

Math

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

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)) -2e-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)) <= -2e-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)) <= (-2d-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)) <= -2e-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)) <= -2e-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)) <= -2e-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)) <= -2e-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], -2e-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))) < -1.999999999999994e-310

   1. Initial program 62.5%

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

   3. Taylor expanded in U around -inf

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

   5. Applied rewrites 29.7%

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

   if -1.999999999999994e-310 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 71.3%

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

   3. Taylor expanded in 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 26.9

         \[\leadsto -U \]

   5. Applied rewrites 26.9%

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

Alternative 13: 13.7% 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 68.9%

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

3. Taylor expanded in U around -inf

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

5. Applied rewrites 26.4%

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

Reproduce

herbie shell --seed 2025064 
(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)))))