Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.1% → 99.4%

Time: 4.9s

Alternatives: 12

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\left(\left(t\_0 \cdot J\_m\right) \cdot -2\right) \cdot \sqrt{{\left(\frac{U\_m}{t\_0 \cdot \left(J\_m + J\_m\right)}\right)}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* -2.0 (* 0.5 U_m))
      (if (<= t_2 5e+297)
        (*
         (* (* t_0 J_m) -2.0)
         (sqrt (+ (pow (/ U_m (* t_0 (+ J_m J_m))) 2.0) 1.0)))
        (* -2.0 (* -0.5 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((0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (0.5 * U_m);
	} else if (t_2 <= 5e+297) {
		tmp = ((t_0 * J_m) * -2.0) * sqrt((pow((U_m / (t_0 * (J_m + J_m))), 2.0) + 1.0));
	} else {
		tmp = -2.0 * (-0.5 * 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((0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (0.5 * U_m);
	} else if (t_2 <= 5e+297) {
		tmp = ((t_0 * J_m) * -2.0) * Math.sqrt((Math.pow((U_m / (t_0 * (J_m + J_m))), 2.0) + 1.0));
	} else {
		tmp = -2.0 * (-0.5 * 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((0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (0.5 * U_m)
	elif t_2 <= 5e+297:
		tmp = ((t_0 * J_m) * -2.0) * math.sqrt((math.pow((U_m / (t_0 * (J_m + J_m))), 2.0) + 1.0))
	else:
		tmp = -2.0 * (-0.5 * 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(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(0.5 * U_m));
	elseif (t_2 <= 5e+297)
		tmp = Float64(Float64(Float64(t_0 * J_m) * -2.0) * sqrt(Float64((Float64(U_m / Float64(t_0 * Float64(J_m + J_m))) ^ 2.0) + 1.0)));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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((0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (0.5 * U_m);
	elseif (t_2 <= 5e+297)
		tmp = ((t_0 * J_m) * -2.0) * sqrt((((U_m / (t_0 * (J_m + J_m))) ^ 2.0) + 1.0));
	else
		tmp = -2.0 * (-0.5 * 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[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+297], N[(N[(N[(t$95$0 * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999998e297

   1. Initial program 73.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.1

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

   3. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.2

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 73.2

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

   7. Applied rewrites 73.2%

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

   if 4.9999999999999998e297 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around -inf

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

       1. lower-*.f64 14.5

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

   12. Applied rewrites 14.5%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot -0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\left(\left(t\_0 \cdot -2\right) \cdot J\_m\right) \cdot \cosh \sinh^{-1} \left(\frac{U\_m}{\left(J\_m + J\_m\right) \cdot t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

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 -0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* -2.0 (* 0.5 U_m))
      (if (<= t_2 5e+297)
        (* (* (* t_0 -2.0) J_m) (cosh (asinh (/ U_m (* (+ J_m J_m) t_0)))))
        (* -2.0 (* -0.5 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 * -0.5));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (0.5 * U_m);
	} else if (t_2 <= 5e+297) {
		tmp = ((t_0 * -2.0) * J_m) * cosh(asinh((U_m / ((J_m + J_m) * t_0))));
	} else {
		tmp = -2.0 * (-0.5 * 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 * -0.5))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (0.5 * U_m)
	elif t_2 <= 5e+297:
		tmp = ((t_0 * -2.0) * J_m) * math.cosh(math.asinh((U_m / ((J_m + J_m) * t_0))))
	else:
		tmp = -2.0 * (-0.5 * 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 * -0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(0.5 * U_m));
	elseif (t_2 <= 5e+297)
		tmp = Float64(Float64(Float64(t_0 * -2.0) * J_m) * cosh(asinh(Float64(U_m / Float64(Float64(J_m + J_m) * t_0)))));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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 * -0.5));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (0.5 * U_m);
	elseif (t_2 <= 5e+297)
		tmp = ((t_0 * -2.0) * J_m) * cosh(asinh((U_m / ((J_m + J_m) * t_0))));
	else
		tmp = -2.0 * (-0.5 * 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 * -0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+297], N[(N[(N[(t$95$0 * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U$95$m / N[(N[(J$95$m + J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999998e297

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 73.2

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

      7. lift-cos.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 73.2

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

      18. lift-sqrt.f64 N/A

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

      19. lift-+.f64 N/A

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

   3. Applied rewrites 84.9%

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

   if 4.9999999999999998e297 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around -inf

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

       1. lower-*.f64 14.5

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

   12. Applied rewrites 14.5%

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

Alternative 3: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(K \cdot -0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot U\_m\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\left(\cosh \sinh^{-1} \left(\frac{U\_m}{\left(J\_m + J\_m\right) \cdot t\_2}\right) \cdot \left(J\_m \cdot -2\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

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)))))
        (t_2 (cos (* K -0.5))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (* -2.0 (* 0.5 U_m))
      (if (<= t_1 5e+297)
        (* (* (cosh (asinh (/ U_m (* (+ J_m J_m) t_2)))) (* J_m -2.0)) t_2)
        (* -2.0 (* -0.5 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 t_2 = cos((K * -0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -2.0 * (0.5 * U_m);
	} else if (t_1 <= 5e+297) {
		tmp = (cosh(asinh((U_m / ((J_m + J_m) * t_2)))) * (J_m * -2.0)) * t_2;
	} else {
		tmp = -2.0 * (-0.5 * 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)))
	t_2 = math.cos((K * -0.5))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -2.0 * (0.5 * U_m)
	elif t_1 <= 5e+297:
		tmp = (math.cosh(math.asinh((U_m / ((J_m + J_m) * t_2)))) * (J_m * -2.0)) * t_2
	else:
		tmp = -2.0 * (-0.5 * 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))))
	t_2 = cos(Float64(K * -0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(0.5 * U_m));
	elseif (t_1 <= 5e+297)
		tmp = Float64(Float64(cosh(asinh(Float64(U_m / Float64(Float64(J_m + J_m) * t_2)))) * Float64(J_m * -2.0)) * t_2);
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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)));
	t_2 = cos((K * -0.5));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -2.0 * (0.5 * U_m);
	elseif (t_1 <= 5e+297)
		tmp = (cosh(asinh((U_m / ((J_m + J_m) * t_2)))) * (J_m * -2.0)) * t_2;
	else
		tmp = -2.0 * (-0.5 * 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]}, Block[{t$95$2 = N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+297], N[(N[(N[Cosh[N[ArcSinh[N[(U$95$m / N[(N[(J$95$m + J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999998e297

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 84.9%

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

   if 4.9999999999999998e297 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around -inf

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

       1. lower-*.f64 14.5

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

   12. Applied rewrites 14.5%

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

Alternative 4: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(K \cdot -0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot U\_m\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\left(\cosh \sinh^{-1} \left(\frac{U\_m}{\left(J\_m + J\_m\right) \cdot t\_2}\right) \cdot t\_2\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

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)))))
        (t_2 (cos (* K -0.5))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (* -2.0 (* 0.5 U_m))
      (if (<= t_1 5e+297)
        (* (* (cosh (asinh (/ U_m (* (+ J_m J_m) t_2)))) t_2) (* J_m -2.0))
        (* -2.0 (* -0.5 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 t_2 = cos((K * -0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -2.0 * (0.5 * U_m);
	} else if (t_1 <= 5e+297) {
		tmp = (cosh(asinh((U_m / ((J_m + J_m) * t_2)))) * t_2) * (J_m * -2.0);
	} else {
		tmp = -2.0 * (-0.5 * 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)))
	t_2 = math.cos((K * -0.5))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -2.0 * (0.5 * U_m)
	elif t_1 <= 5e+297:
		tmp = (math.cosh(math.asinh((U_m / ((J_m + J_m) * t_2)))) * t_2) * (J_m * -2.0)
	else:
		tmp = -2.0 * (-0.5 * 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))))
	t_2 = cos(Float64(K * -0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(0.5 * U_m));
	elseif (t_1 <= 5e+297)
		tmp = Float64(Float64(cosh(asinh(Float64(U_m / Float64(Float64(J_m + J_m) * t_2)))) * t_2) * Float64(J_m * -2.0));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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)));
	t_2 = cos((K * -0.5));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -2.0 * (0.5 * U_m);
	elseif (t_1 <= 5e+297)
		tmp = (cosh(asinh((U_m / ((J_m + J_m) * t_2)))) * t_2) * (J_m * -2.0);
	else
		tmp = -2.0 * (-0.5 * 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]}, Block[{t$95$2 = N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+297], N[(N[(N[Cosh[N[ArcSinh[N[(U$95$m / N[(N[(J$95$m + J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999998e297

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 84.9%

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

   if 4.9999999999999998e297 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around -inf

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

       1. lower-*.f64 14.5

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

   12. Applied rewrites 14.5%

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

Alternative 5: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U\_m}{J\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

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))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* -2.0 (* 0.5 U_m))
      (if (<= t_2 2e-14)
        (* t_1 (sqrt (+ 1.0 (pow (* 0.5 (/ U_m J_m)) 2.0))))
        (* -2.0 (* -0.5 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 = -2.0 * (0.5 * U_m);
	} else if (t_2 <= 2e-14) {
		tmp = t_1 * sqrt((1.0 + pow((0.5 * (U_m / J_m)), 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * 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 = -2.0 * (0.5 * U_m);
	} else if (t_2 <= 2e-14) {
		tmp = t_1 * Math.sqrt((1.0 + Math.pow((0.5 * (U_m / J_m)), 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * 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 = -2.0 * (0.5 * U_m)
	elif t_2 <= 2e-14:
		tmp = t_1 * math.sqrt((1.0 + math.pow((0.5 * (U_m / J_m)), 2.0)))
	else:
		tmp = -2.0 * (-0.5 * 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(-2.0 * Float64(0.5 * U_m));
	elseif (t_2 <= 2e-14)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U_m / J_m)) ^ 2.0))));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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 = -2.0 * (0.5 * U_m);
	elseif (t_2 <= 2e-14)
		tmp = t_1 * sqrt((1.0 + ((0.5 * (U_m / J_m)) ^ 2.0)));
	else
		tmp = -2.0 * (-0.5 * 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)], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-14], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2e-14

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 63.9

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

   4. Applied rewrites 63.9%

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

   if 2e-14 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around -inf

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

       1. lower-*.f64 14.5

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

   12. Applied rewrites 14.5%

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

Alternative 6: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot U\_m\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\cosh \sinh^{-1} \left(0.5 \cdot \frac{U\_m}{J\_m}\right) \cdot \cos \left(K \cdot -0.5\right)\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

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))
      (* -2.0 (* 0.5 U_m))
      (if (<= t_1 2e-14)
        (*
         (* (cosh (asinh (* 0.5 (/ U_m J_m)))) (cos (* K -0.5)))
         (* J_m -2.0))
        (* -2.0 (* -0.5 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 = -2.0 * (0.5 * U_m);
	} else if (t_1 <= 2e-14) {
		tmp = (cosh(asinh((0.5 * (U_m / J_m)))) * cos((K * -0.5))) * (J_m * -2.0);
	} else {
		tmp = -2.0 * (-0.5 * 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 <= -math.inf:
		tmp = -2.0 * (0.5 * U_m)
	elif t_1 <= 2e-14:
		tmp = (math.cosh(math.asinh((0.5 * (U_m / J_m)))) * math.cos((K * -0.5))) * (J_m * -2.0)
	else:
		tmp = -2.0 * (-0.5 * 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(-2.0 * Float64(0.5 * U_m));
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(cosh(asinh(Float64(0.5 * Float64(U_m / J_m)))) * cos(Float64(K * -0.5))) * Float64(J_m * -2.0));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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 <= -Inf)
		tmp = -2.0 * (0.5 * U_m);
	elseif (t_1 <= 2e-14)
		tmp = (cosh(asinh((0.5 * (U_m / J_m)))) * cos((K * -0.5))) * (J_m * -2.0);
	else
		tmp = -2.0 * (-0.5 * 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, (-Infinity)], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(N[Cosh[N[ArcSinh[N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2e-14

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.1

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

   6. Applied rewrites 71.1%

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

   if 2e-14 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around -inf

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

       1. lower-*.f64 14.5

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

   12. Applied rewrites 14.5%

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

Alternative 7: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot U\_m\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(\cos \left(K \cdot -0.5\right) \cdot -2\right) \cdot J\_m\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

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))
      (* -2.0 (* 0.5 U_m))
      (if (<= t_1 2e-14)
        (*
         (* (* (cos (* K -0.5)) -2.0) J_m)
         (cosh (asinh (* 0.5 (/ U_m J_m)))))
        (* -2.0 (* -0.5 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 = -2.0 * (0.5 * U_m);
	} else if (t_1 <= 2e-14) {
		tmp = ((cos((K * -0.5)) * -2.0) * J_m) * cosh(asinh((0.5 * (U_m / J_m))));
	} else {
		tmp = -2.0 * (-0.5 * 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 <= -math.inf:
		tmp = -2.0 * (0.5 * U_m)
	elif t_1 <= 2e-14:
		tmp = ((math.cos((K * -0.5)) * -2.0) * J_m) * math.cosh(math.asinh((0.5 * (U_m / J_m))))
	else:
		tmp = -2.0 * (-0.5 * 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(-2.0 * Float64(0.5 * U_m));
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(Float64(cos(Float64(K * -0.5)) * -2.0) * J_m) * cosh(asinh(Float64(0.5 * Float64(U_m / J_m)))));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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 <= -Inf)
		tmp = -2.0 * (0.5 * U_m);
	elseif (t_1 <= 2e-14)
		tmp = ((cos((K * -0.5)) * -2.0) * J_m) * cosh(asinh((0.5 * (U_m / J_m))));
	else
		tmp = -2.0 * (-0.5 * 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, (-Infinity)], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(N[(N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2e-14

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 73.2

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

      7. lift-cos.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 73.2

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

      18. lift-sqrt.f64 N/A

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

      19. lift-+.f64 N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.2

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

   6. Applied rewrites 71.2%

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

   if 2e-14 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around -inf

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

       1. lower-*.f64 14.5

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

   12. Applied rewrites 14.5%

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

Alternative 8: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot U\_m\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\cos \left(-0.5 \cdot K\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-277}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U\_m}{J\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

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))
      (* -2.0 (* 0.5 U_m))
      (if (<= t_1 -5e+93)
        (* (cos (* -0.5 K)) (* J_m -2.0))
        (if (<= t_1 -5e-277)
          (* (* -2.0 J_m) (cosh (asinh (* 0.5 (/ U_m J_m)))))
          (* -2.0 (* -0.5 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 = -2.0 * (0.5 * U_m);
	} else if (t_1 <= -5e+93) {
		tmp = cos((-0.5 * K)) * (J_m * -2.0);
	} else if (t_1 <= -5e-277) {
		tmp = (-2.0 * J_m) * cosh(asinh((0.5 * (U_m / J_m))));
	} else {
		tmp = -2.0 * (-0.5 * 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 <= -math.inf:
		tmp = -2.0 * (0.5 * U_m)
	elif t_1 <= -5e+93:
		tmp = math.cos((-0.5 * K)) * (J_m * -2.0)
	elif t_1 <= -5e-277:
		tmp = (-2.0 * J_m) * math.cosh(math.asinh((0.5 * (U_m / J_m))))
	else:
		tmp = -2.0 * (-0.5 * 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(-2.0 * Float64(0.5 * U_m));
	elseif (t_1 <= -5e+93)
		tmp = Float64(cos(Float64(-0.5 * K)) * Float64(J_m * -2.0));
	elseif (t_1 <= -5e-277)
		tmp = Float64(Float64(-2.0 * J_m) * cosh(asinh(Float64(0.5 * Float64(U_m / J_m)))));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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 <= -Inf)
		tmp = -2.0 * (0.5 * U_m);
	elseif (t_1 <= -5e+93)
		tmp = cos((-0.5 * K)) * (J_m * -2.0);
	elseif (t_1 <= -5e-277)
		tmp = (-2.0 * J_m) * cosh(asinh((0.5 * (U_m / J_m))));
	else
		tmp = -2.0 * (-0.5 * 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, (-Infinity)], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+93], N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-277], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000001e93

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in J around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 51.9

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

   6. Applied rewrites 51.9%

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

   if -5.0000000000000001e93 < (*.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))))) < -5e-277

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 73.2

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

      7. lift-cos.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 73.2

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

      18. lift-sqrt.f64 N/A

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

      19. lift-+.f64 N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.2

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

   6. Applied rewrites 71.2%

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

   7. Taylor expanded in K around 0

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

   9. Applied rewrites 53.3%

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

   if -5e-277 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around -inf

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

       1. lower-*.f64 14.5

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

   12. Applied rewrites 14.5%

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

Alternative 9: 76.5% accurate, 0.4× speedup

Math

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

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))
      (* -2.0 (* 0.5 U_m))
      (if (<= t_1 -5e-277)
        (* (* -2.0 J_m) (cosh (asinh (* 0.5 (/ U_m J_m)))))
        (* -2.0 (* -0.5 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 = -2.0 * (0.5 * U_m);
	} else if (t_1 <= -5e-277) {
		tmp = (-2.0 * J_m) * cosh(asinh((0.5 * (U_m / J_m))));
	} else {
		tmp = -2.0 * (-0.5 * 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 <= -math.inf:
		tmp = -2.0 * (0.5 * U_m)
	elif t_1 <= -5e-277:
		tmp = (-2.0 * J_m) * math.cosh(math.asinh((0.5 * (U_m / J_m))))
	else:
		tmp = -2.0 * (-0.5 * 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(-2.0 * Float64(0.5 * U_m));
	elseif (t_1 <= -5e-277)
		tmp = Float64(Float64(-2.0 * J_m) * cosh(asinh(Float64(0.5 * Float64(U_m / J_m)))));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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 <= -Inf)
		tmp = -2.0 * (0.5 * U_m);
	elseif (t_1 <= -5e-277)
		tmp = (-2.0 * J_m) * cosh(asinh((0.5 * (U_m / J_m))));
	else
		tmp = -2.0 * (-0.5 * 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, (-Infinity)], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-277], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(0.5 * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5e-277

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 73.2

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

      7. lift-cos.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 73.2

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

      18. lift-sqrt.f64 N/A

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

      19. lift-+.f64 N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.2

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

   6. Applied rewrites 71.2%

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

   7. Taylor expanded in K around 0

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

   9. Applied rewrites 53.3%

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

   if -5e-277 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around -inf

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

       1. lower-*.f64 14.5

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

   12. Applied rewrites 14.5%

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

Alternative 10: 60.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := -2 \cdot \left(0.5 \cdot U\_m\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(-2, J\_m, -0.25 \cdot \frac{{U\_m}^{2}}{J\_m}\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-277}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 U_m)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      t_0
      (if (<= t_2 -1e+62)
        (fma -2.0 J_m (* -0.25 (/ (pow U_m 2.0) J_m)))
        (if (<= t_2 -5e-277) t_0 (* -2.0 (* -0.5 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 = -2.0 * (0.5 * U_m);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= -1e+62) {
		tmp = fma(-2.0, J_m, (-0.25 * (pow(U_m, 2.0) / J_m)));
	} else if (t_2 <= -5e-277) {
		tmp = t_0;
	} else {
		tmp = -2.0 * (-0.5 * 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(-2.0 * Float64(0.5 * U_m))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= -1e+62)
		tmp = fma(-2.0, J_m, Float64(-0.25 * Float64((U_m ^ 2.0) / J_m)));
	elseif (t_2 <= -5e-277)
		tmp = t_0;
	else
		tmp = Float64(-2.0 * Float64(-0.5 * 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[(-2.0 * N[(0.5 * U$95$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]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, -1e+62], N[(-2.0 * J$95$m + N[(-0.25 * N[(N[Power[U$95$m, 2.0], $MachinePrecision] / J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-277], t$95$0, N[(-2.0 * N[(-0.5 * U$95$m), $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 or -1.00000000000000004e62 < (*.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))))) < -5e-277

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 22.3

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 22.3

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

   9. Applied rewrites 22.3%

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

   10. Taylor expanded in U around 0

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

       1. lower-*.f64 38.9

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

   12. Applied rewrites 38.9%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.00000000000000004e62

   1. Initial program 73.1%

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

   2. Taylor expanded in U around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 28.6

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

   7. Applied rewrites 28.6%

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

   if -5e-277 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.1%

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

   2. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 29.6

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

   4. Applied rewrites 29.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      3. sqrt-prod N/A

         \[\leadsto -2 \cdot \left(\sqrt{\frac{1}{4}} \cdot \sqrt{{U}^{2}}\right) \]

      4. metadata-eval N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

      6. lower-sqrt.f64 22.3

         \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{{U}^{2}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

      8. unpow2 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{U \cdot U}\right) \]

      9. lower-*.f64 22.3

         \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{U \cdot U}\right) \]

   9. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{U \cdot U}\right) \]

   10. Taylor expanded in U around -inf

       \[\leadsto -2 \cdot \left(\frac{-1}{2} \cdot U\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 14.5

          \[\leadsto -2 \cdot \left(-0.5 \cdot U\right) \]

   12. Applied rewrites 14.5%

       \[\leadsto -2 \cdot \left(-0.5 \cdot U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(0.5 \cdot U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (*
  J_s
  (if (<= (cos (/ K 2.0)) -4e-310)
    (* -2.0 (* -0.5 U_m))
    (* -2.0 (* 0.5 U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (cos((K / 2.0)) <= -4e-310) {
		tmp = -2.0 * (-0.5 * U_m);
	} else {
		tmp = -2.0 * (0.5 * U_m);
	}
	return J_s * tmp;
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-4d-310)) then
        tmp = (-2.0d0) * ((-0.5d0) * u_m)
    else
        tmp = (-2.0d0) * (0.5d0 * u_m)
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -4e-310) {
		tmp = -2.0 * (-0.5 * U_m);
	} else {
		tmp = -2.0 * (0.5 * U_m);
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if math.cos((K / 2.0)) <= -4e-310:
		tmp = -2.0 * (-0.5 * U_m)
	else:
		tmp = -2.0 * (0.5 * U_m)
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -4e-310)
		tmp = Float64(-2.0 * Float64(-0.5 * U_m));
	else
		tmp = Float64(-2.0 * Float64(0.5 * U_m));
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -4e-310)
		tmp = -2.0 * (-0.5 * U_m);
	else
		tmp = -2.0 * (0.5 * U_m);
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -4e-310], N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -3.999999999999988e-310

   1. Initial program 73.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4}} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      11. lower-*.f64 29.6

          \[\leadsto -2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 29.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      3. sqrt-prod N/A

         \[\leadsto -2 \cdot \left(\sqrt{\frac{1}{4}} \cdot \sqrt{{U}^{2}}\right) \]

      4. metadata-eval N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

      6. lower-sqrt.f64 22.3

         \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{{U}^{2}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

      8. unpow2 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{U \cdot U}\right) \]

      9. lower-*.f64 22.3

         \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{U \cdot U}\right) \]

   9. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{U \cdot U}\right) \]

   10. Taylor expanded in U around -inf

       \[\leadsto -2 \cdot \left(\frac{-1}{2} \cdot U\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 14.5

          \[\leadsto -2 \cdot \left(-0.5 \cdot U\right) \]

   12. Applied rewrites 14.5%

       \[\leadsto -2 \cdot \left(-0.5 \cdot U\right) \]

   if -3.999999999999988e-310 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 73.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4}} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      11. lower-*.f64 29.6

          \[\leadsto -2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 29.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      3. lower-pow.f64 22.3

         \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

   7. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

      3. sqrt-prod N/A

         \[\leadsto -2 \cdot \left(\sqrt{\frac{1}{4}} \cdot \sqrt{{U}^{2}}\right) \]

      4. metadata-eval N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

      6. lower-sqrt.f64 22.3

         \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{{U}^{2}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

      8. unpow2 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{U \cdot U}\right) \]

      9. lower-*.f64 22.3

         \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{U \cdot U}\right) \]

   9. Applied rewrites 22.3%

      \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{U \cdot U}\right) \]

   10. Taylor expanded in U around 0

       \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot U\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 38.9

          \[\leadsto -2 \cdot \left(0.5 \cdot U\right) \]

   12. Applied rewrites 38.9%

       \[\leadsto -2 \cdot \left(0.5 \cdot U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 14.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(-2 \cdot \left(-0.5 \cdot U\_m\right)\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (* -2.0 (* -0.5 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 * (-2.0 * (-0.5 * 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 * ((-2.0d0) * ((-0.5d0) * 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 * (-2.0 * (-0.5 * 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 * (-2.0 * (-0.5 * U_m))

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(-2.0 * Float64(-0.5 * 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 * (-2.0 * (-0.5 * 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 * N[(-2.0 * N[(-0.5 * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

2. Taylor expanded in J around 0

   \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -2 \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

   3. lower-cos.f64 N/A

      \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4}} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

   7. lower-/.f64 N/A

      \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

   11. lower-*.f64 29.6

       \[\leadsto -2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

4. Applied rewrites 29.6%

   \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right)} \]

5. Taylor expanded in K around 0

   \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]
6. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

   3. lower-pow.f64 22.3

      \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]

7. Applied rewrites 22.3%

   \[\leadsto -2 \cdot \sqrt{0.25 \cdot {U}^{2}} \]
8. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto -2 \cdot \sqrt{\frac{1}{4} \cdot {U}^{2}} \]

   3. sqrt-prod N/A

      \[\leadsto -2 \cdot \left(\sqrt{\frac{1}{4}} \cdot \sqrt{{U}^{2}}\right) \]

   4. metadata-eval N/A

      \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

   6. lower-sqrt.f64 22.3

      \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{{U}^{2}}\right) \]

   7. lift-pow.f64 N/A

      \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{{U}^{2}}\right) \]

   8. unpow2 N/A

      \[\leadsto -2 \cdot \left(\frac{1}{2} \cdot \sqrt{U \cdot U}\right) \]

   9. lower-*.f64 22.3

      \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{U \cdot U}\right) \]

9. Applied rewrites 22.3%

   \[\leadsto -2 \cdot \left(0.5 \cdot \sqrt{U \cdot U}\right) \]

10. Taylor expanded in U around -inf

    \[\leadsto -2 \cdot \left(\frac{-1}{2} \cdot U\right) \]
11. Step-by-step derivation

    1. lower-*.f64 14.5

       \[\leadsto -2 \cdot \left(-0.5 \cdot U\right) \]

12. Applied rewrites 14.5%

    \[\leadsto -2 \cdot \left(-0.5 \cdot U\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025148 
(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)))))