Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.6% → 99.7%

Time: 6.3s

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: 73.6% 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: 99.7% accurate, 0.3× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_2 5e+304)
        (*
         (* (* -2.0 J_m) t_0)
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
        (* (* -0.5 U_m) -2.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_2 <= 5e+304) {
		tmp = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_2 <= 5e+304) {
		tmp = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K * 0.5))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (0.5 * U_m) * -2.0
	elif t_2 <= 5e+304:
		tmp = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	else:
		tmp = (-0.5 * U_m) * -2.0
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_2 <= 5e+304)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))));
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K * 0.5));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (0.5 * U_m) * -2.0;
	elseif (t_2 <= 5e+304)
		tmp = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	else
		tmp = (-0.5 * U_m) * -2.0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999997e304

   1. Initial program 73.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 73.6

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

   3. Applied rewrites 73.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 73.6

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

   5. Applied rewrites 73.6%

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

   if 4.9999999999999997e304 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\left(\left(t\_0 \cdot J\_m\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U\_m}{t\_0 \cdot \left(J\_m + J\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_2 5e+304)
        (* (* (* t_0 J_m) -2.0) (cosh (asinh (/ U_m (* t_0 (+ J_m J_m))))))
        (* (* -0.5 U_m) -2.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_2 <= 5e+304) {
		tmp = ((t_0 * J_m) * -2.0) * cosh(asinh((U_m / (t_0 * (J_m + J_m)))));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (0.5 * U_m) * -2.0
	elif t_2 <= 5e+304:
		tmp = ((t_0 * J_m) * -2.0) * math.cosh(math.asinh((U_m / (t_0 * (J_m + J_m)))))
	else:
		tmp = (-0.5 * U_m) * -2.0
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_2 <= 5e+304)
		tmp = Float64(Float64(Float64(t_0 * J_m) * -2.0) * cosh(asinh(Float64(U_m / Float64(t_0 * Float64(J_m + J_m))))));
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (0.5 * U_m) * -2.0;
	elseif (t_2 <= 5e+304)
		tmp = ((t_0 * J_m) * -2.0) * cosh(asinh((U_m / (t_0 * (J_m + J_m)))));
	else
		tmp = (-0.5 * U_m) * -2.0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 5e+304], N[(N[(N[(t$95$0 * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U$95$m / N[(t$95$0 * N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999997e304

   1. Initial program 73.6%

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

   3. Applied rewrites 85.0%

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

   if 4.9999999999999997e304 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 3: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\left(J\_m \cdot -2\right) \cdot \left(t\_0 \cdot \cosh \sinh^{-1} \left(\frac{U\_m}{t\_0 \cdot \left(J\_m + J\_m\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_2 5e+304)
        (* (* J_m -2.0) (* t_0 (cosh (asinh (/ U_m (* t_0 (+ J_m J_m)))))))
        (* (* -0.5 U_m) -2.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_2 <= 5e+304) {
		tmp = (J_m * -2.0) * (t_0 * cosh(asinh((U_m / (t_0 * (J_m + J_m))))));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J_m) * t_1) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (0.5 * U_m) * -2.0
	elif t_2 <= 5e+304:
		tmp = (J_m * -2.0) * (t_0 * math.cosh(math.asinh((U_m / (t_0 * (J_m + J_m))))))
	else:
		tmp = (-0.5 * U_m) * -2.0
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_2 <= 5e+304)
		tmp = Float64(Float64(J_m * -2.0) * Float64(t_0 * cosh(asinh(Float64(U_m / Float64(t_0 * Float64(J_m + J_m)))))));
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (0.5 * U_m) * -2.0;
	elseif (t_2 <= 5e+304)
		tmp = (J_m * -2.0) * (t_0 * cosh(asinh((U_m / (t_0 * (J_m + J_m))))));
	else
		tmp = (-0.5 * U_m) * -2.0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 5e+304], N[(N[(J$95$m * -2.0), $MachinePrecision] * N[(t$95$0 * N[Cosh[N[ArcSinh[N[(U$95$m / N[(t$95$0 * N[(J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999997e304

   1. Initial program 73.6%

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

   3. Applied rewrites 85.0%

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

   if 4.9999999999999997e304 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 4: 91.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = 0.5 * (U_m / J_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_1 <= 5e+304) {
		tmp = ((cos((0.5 * K)) * J_m) * -2.0) * sqrt(fma(t_2, t_2, 1.0));
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = Float64(0.5 * Float64(U_m / J_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_1 <= 5e+304)
		tmp = Float64(Float64(Float64(cos(Float64(0.5 * K)) * J_m) * -2.0) * sqrt(fma(t_2, t_2, 1.0)));
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999997e304

   1. Initial program 73.6%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.9

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

   6. Applied rewrites 71.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 65.1

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

   8. Applied rewrites 65.1%

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

   if 4.9999999999999997e304 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 5: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot 1\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\_m\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{+177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-55}:\\ \;\;\;\;t\_2 \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (* (* (cos (* 0.5 K)) J_m) -2.0) 1.0))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J_m) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_3 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_3 -5e+177)
        t_0
        (if (<= t_3 -4e-55)
          (* t_2 (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)))
          (if (<= t_3 -2e-243)
            (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
            (if (<= t_3 5e+304) t_0 (* (* -0.5 U_m) -2.0)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = ((cos((0.5 * K)) * J_m) * -2.0) * 1.0;
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J_m) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_3 <= -5e+177) {
		tmp = t_0;
	} else if (t_3 <= -4e-55) {
		tmp = t_2 * sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0));
	} else if (t_3 <= -2e-243) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_3 <= 5e+304) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(Float64(cos(Float64(0.5 * K)) * J_m) * -2.0) * 1.0)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J_m) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_3 <= -5e+177)
		tmp = t_0;
	elseif (t_3 <= -4e-55)
		tmp = Float64(t_2 * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)));
	elseif (t_3 <= -2e-243)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0);
	elseif (t_3 <= 5e+304)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 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$3, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$3, -5e+177], t$95$0, If[LessEqual[t$95$3, -4e-55], N[(t$95$2 * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-243], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$3, 5e+304], t$95$0, N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000003e177 or -1.99999999999999999e-243 < (*.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))))) < 4.9999999999999997e304

   1. Initial program 73.6%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in J around inf

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

   6. Applied rewrites 52.6%

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

   if -5.0000000000000003e177 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999998e-55

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

   if -3.99999999999999998e-55 < (*.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))))) < -1.99999999999999999e-243

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 45.5

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

   6. Applied rewrites 45.5%

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

   if 4.9999999999999997e304 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 6: 83.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot 1\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-243}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (* (* (cos (* 0.5 K)) J_m) -2.0) 1.0))
        (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))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_2 -5e+22)
        t_0
        (if (<= t_2 -2e-243)
          (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
          (if (<= t_2 5e+304) t_0 (* (* -0.5 U_m) -2.0))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = ((cos((0.5 * K)) * J_m) * -2.0) * 1.0;
	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 = (0.5 * U_m) * -2.0;
	} else if (t_2 <= -5e+22) {
		tmp = t_0;
	} else if (t_2 <= -2e-243) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_2 <= 5e+304) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(Float64(cos(Float64(0.5 * K)) * J_m) * -2.0) * 1.0)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_2 <= -5e+22)
		tmp = t_0;
	elseif (t_2 <= -2e-243)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0);
	elseif (t_2 <= 5e+304)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, -5e+22], t$95$0, If[LessEqual[t$95$2, -2e-243], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, 5e+304], t$95$0, N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999996e22 or -1.99999999999999999e-243 < (*.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))))) < 4.9999999999999997e304

   1. Initial program 73.6%

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in J around inf

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

   6. Applied rewrites 52.6%

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

   if -4.9999999999999996e22 < (*.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))))) < -1.99999999999999999e-243

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 45.5

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

   6. Applied rewrites 45.5%

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

   if 4.9999999999999997e304 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 7: 77.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot U\_m\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (* (* 0.5 U_m) -2.0)
      (if (<= t_1 -2e-306)
        (* (* (sqrt (fma (* (/ U_m J_m) (/ U_m J_m)) 0.25 1.0)) J_m) -2.0)
        (* (* -0.5 U_m) -2.0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (0.5 * U_m) * -2.0;
	} else if (t_1 <= -2e-306) {
		tmp = (sqrt(fma(((U_m / J_m) * (U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * U_m) * -2.0);
	elseif (t_1 <= -2e-306)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), 0.25, 1.0)) * J_m) * -2.0);
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -2e-306], N[(N[(N[Sqrt[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000006e-306

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 45.5

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

   6. Applied rewrites 45.5%

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

   if -2.00000000000000006e-306 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 8: 69.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot U\_m\right) \cdot -2\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+177}:\\ \;\;\;\;J\_m \cdot \mathsf{fma}\left(-0.25, \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, -2\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-121}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 0.25, 1\right)} \cdot J\_m\right) \cdot -2\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 U_m) -2.0))
        (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 -5e+177)
        (* J_m (fma -0.25 (* (/ U_m J_m) (/ U_m J_m)) -2.0))
        (if (<= t_2 -2e-121)
          (* (* (sqrt (fma (/ (* U_m U_m) (* J_m J_m)) 0.25 1.0)) J_m) -2.0)
          (if (<= t_2 -2e-306) t_0 (* (* -0.5 U_m) -2.0))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = (0.5 * U_m) * -2.0;
	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 <= -5e+177) {
		tmp = J_m * fma(-0.25, ((U_m / J_m) * (U_m / J_m)), -2.0);
	} else if (t_2 <= -2e-121) {
		tmp = (sqrt(fma(((U_m * U_m) / (J_m * J_m)), 0.25, 1.0)) * J_m) * -2.0;
	} else if (t_2 <= -2e-306) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(0.5 * U_m) * -2.0)
	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 <= -5e+177)
		tmp = Float64(J_m * fma(-0.25, Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), -2.0));
	elseif (t_2 <= -2e-121)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 0.25, 1.0)) * J_m) * -2.0);
	elseif (t_2 <= -2e-306)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $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, -5e+177], N[(J$95$m * N[(-0.25 * N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-121], N[(N[(N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$2, -2e-306], t$95$0, N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or -2e-121 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000006e-306

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.0000000000000003e177

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. sub-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 26.7

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

   10. Applied rewrites 26.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-/.f64 30.5

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

   12. Applied rewrites 30.5%

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

   if -5.0000000000000003e177 < (*.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-121

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   if -2.00000000000000006e-306 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 9: 63.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot U\_m\right) \cdot -2\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-121}:\\ \;\;\;\;J\_m \cdot \mathsf{fma}\left(-0.25, \frac{U\_m}{J\_m} \cdot \frac{U\_m}{J\_m}, -2\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 U_m) -2.0))
        (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 -2e-121)
        (* J_m (fma -0.25 (* (/ U_m J_m) (/ U_m J_m)) -2.0))
        (if (<= t_2 -2e-306) t_0 (* (* -0.5 U_m) -2.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = (0.5 * U_m) * -2.0;
	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 <= -2e-121) {
		tmp = J_m * fma(-0.25, ((U_m / J_m) * (U_m / J_m)), -2.0);
	} else if (t_2 <= -2e-306) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(0.5 * U_m) * -2.0)
	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 <= -2e-121)
		tmp = Float64(J_m * fma(-0.25, Float64(Float64(U_m / J_m) * Float64(U_m / J_m)), -2.0));
	elseif (t_2 <= -2e-306)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $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, -2e-121], N[(J$95$m * N[(-0.25 * N[(N[(U$95$m / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-306], t$95$0, N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or -2e-121 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000006e-306

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2e-121

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. sub-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 26.7

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

   10. Applied rewrites 26.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-/.f64 30.5

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

   12. Applied rewrites 30.5%

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

   if -2.00000000000000006e-306 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 10: 63.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot U\_m\right) \cdot -2\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-121}:\\ \;\;\;\;J\_m \cdot -2\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-306}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot U\_m\right) \cdot -2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 U_m) -2.0))
        (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 -2e-121)
        (* J_m -2.0)
        (if (<= t_2 -2e-306) t_0 (* (* -0.5 U_m) -2.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = (0.5 * U_m) * -2.0;
	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 <= -2e-121) {
		tmp = J_m * -2.0;
	} else if (t_2 <= -2e-306) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = (0.5 * U_m) * -2.0;
	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 = t_0;
	} else if (t_2 <= -2e-121) {
		tmp = J_m * -2.0;
	} else if (t_2 <= -2e-306) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * U_m) * -2.0;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = (0.5 * U_m) * -2.0
	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 = t_0
	elif t_2 <= -2e-121:
		tmp = J_m * -2.0
	elif t_2 <= -2e-306:
		tmp = t_0
	else:
		tmp = (-0.5 * U_m) * -2.0
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(0.5 * U_m) * -2.0)
	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 <= -2e-121)
		tmp = Float64(J_m * -2.0);
	elseif (t_2 <= -2e-306)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * U_m) * -2.0);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = (0.5 * U_m) * -2.0;
	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 = t_0;
	elseif (t_2 <= -2e-121)
		tmp = J_m * -2.0;
	elseif (t_2 <= -2e-306)
		tmp = t_0;
	else
		tmp = (-0.5 * U_m) * -2.0;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(0.5 * U$95$m), $MachinePrecision] * -2.0), $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, -2e-121], N[(J$95$m * -2.0), $MachinePrecision], If[LessEqual[t$95$2, -2e-306], t$95$0, N[(N[(-0.5 * U$95$m), $MachinePrecision] * -2.0), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or -2e-121 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000006e-306

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around 0

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

      1. lower-*.f64 38.8

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

   10. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2e-121

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. sub-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 26.7

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

   10. Applied rewrites 26.7%

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

   11. Taylor expanded in J around inf

       \[\leadsto J \cdot -2 \]
   12. Step-by-step derivation

   13. Applied rewrites 29.9%

       \[\leadsto J \cdot -2 \]

   if -2.00000000000000006e-306 < (*.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 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

Alternative 11: 42.3% accurate, 2.4× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= (cos (/ K 2.0)) -5e-310) (* (* -0.5 U_m) -2.0) (* J_m -2.0))))

C

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

Fortran

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

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

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-5d-310)) then
        tmp = ((-0.5d0) * u_m) * (-2.0d0)
    else
        tmp = j_m * (-2.0d0)
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -5e-310) {
		tmp = (-0.5 * U_m) * -2.0;
	} else {
		tmp = J_m * -2.0;
	}
	return J_s * tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 33.0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in J around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in U around -inf

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

      1. lower-*.f64 13.9

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

   10. Applied rewrites 13.9%

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

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

   1. Initial program 73.6%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{{J}^{2}}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      12. unpow2 N/A

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

      13. lower-*.f64 33.0

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2 \]

   4. Applied rewrites 33.0%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]

   5. Taylor expanded in J around 0

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

      3. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

      4. lift-*.f64 21.8

         \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   7. Applied rewrites 21.8%

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

   8. Taylor expanded in J around inf

      \[\leadsto J \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} - 2\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto J \cdot \left(\frac{-1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} - \color{blue}{2}\right) \]

      2. sub-flip N/A

         \[\leadsto J \cdot \left(\frac{-1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} + \left(\mathsf{neg}\left(2\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto J \cdot \left(\frac{-1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} + -2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{U}^{2}}{\color{blue}{{J}^{2}}}, -2\right) \]

      5. lower-/.f64 N/A

         \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{U}^{2}}{{J}^{\color{blue}{2}}}, -2\right) \]

      6. pow2 N/A

         \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{U \cdot U}{{J}^{2}}, -2\right) \]

      7. lift-*.f64 N/A

         \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{U \cdot U}{{J}^{2}}, -2\right) \]

      8. pow2 N/A

         \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{U \cdot U}{J \cdot J}, -2\right) \]

      9. lower-*.f64 26.7

         \[\leadsto J \cdot \mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot J}, -2\right) \]

   10. Applied rewrites 26.7%

       \[\leadsto J \cdot \color{blue}{\mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot J}, -2\right)} \]

   11. Taylor expanded in J around inf

       \[\leadsto J \cdot -2 \]
   12. Step-by-step derivation

   13. Applied rewrites 29.9%

       \[\leadsto J \cdot -2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 29.9% accurate, 27.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(J\_m \cdot -2\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (* J_m -2.0)))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (J_m * -2.0);
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * (j_m * (-2.0d0))
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * (J_m * -2.0);
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * (J_m * -2.0)

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(J_m * -2.0))
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * (J_m * -2.0);
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

2. Taylor expanded in K around 0

   \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \cdot \color{blue}{-2} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \cdot \color{blue}{-2} \]

   3. *-commutative N/A

      \[\leadsto \left(\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right) \cdot -2 \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right) \cdot -2 \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot J\right) \cdot -2 \]

   6. +-commutative N/A

      \[\leadsto \left(\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} + 1} \cdot J\right) \cdot -2 \]

   7. *-commutative N/A

      \[\leadsto \left(\sqrt{\frac{{U}^{2}}{{J}^{2}} \cdot \frac{1}{4} + 1} \cdot J\right) \cdot -2 \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{{U}^{2}}{{J}^{2}}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

   9. lower-/.f64 N/A

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{{U}^{2}}{{J}^{2}}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

   10. unpow2 N/A

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{{J}^{2}}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{{J}^{2}}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

   12. unpow2 N/A

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, \frac{1}{4}, 1\right)} \cdot J\right) \cdot -2 \]

   13. lower-*.f64 33.0

       \[\leadsto \left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2 \]

4. Applied rewrites 33.0%

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2} \]

5. Taylor expanded in J around 0

   \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]
6. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{4} \cdot {U}^{2}} \cdot -2 \]

   3. pow2 N/A

      \[\leadsto \sqrt{\frac{1}{4} \cdot \left(U \cdot U\right)} \cdot -2 \]

   4. lift-*.f64 21.8

      \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

7. Applied rewrites 21.8%

   \[\leadsto \sqrt{0.25 \cdot \left(U \cdot U\right)} \cdot -2 \]

8. Taylor expanded in J around inf

   \[\leadsto J \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} - 2\right)} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto J \cdot \left(\frac{-1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} - \color{blue}{2}\right) \]

   2. sub-flip N/A

      \[\leadsto J \cdot \left(\frac{-1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} + \left(\mathsf{neg}\left(2\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto J \cdot \left(\frac{-1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} + -2\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{U}^{2}}{\color{blue}{{J}^{2}}}, -2\right) \]

   5. lower-/.f64 N/A

      \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{{U}^{2}}{{J}^{\color{blue}{2}}}, -2\right) \]

   6. pow2 N/A

      \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{U \cdot U}{{J}^{2}}, -2\right) \]

   7. lift-*.f64 N/A

      \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{U \cdot U}{{J}^{2}}, -2\right) \]

   8. pow2 N/A

      \[\leadsto J \cdot \mathsf{fma}\left(\frac{-1}{4}, \frac{U \cdot U}{J \cdot J}, -2\right) \]

   9. lower-*.f64 26.7

      \[\leadsto J \cdot \mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot J}, -2\right) \]

10. Applied rewrites 26.7%

    \[\leadsto J \cdot \color{blue}{\mathsf{fma}\left(-0.25, \frac{U \cdot U}{J \cdot J}, -2\right)} \]

11. Taylor expanded in J around inf

    \[\leadsto J \cdot -2 \]
12. Step-by-step derivation

13. Applied rewrites 29.9%

    \[\leadsto J \cdot -2 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025135 
(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)))))