Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.7% → 85.2%

Time: 7.8s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

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: 72.7% accurate, 1.0× speedup

Math

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

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: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|J\right| + \left|J\right|\\ t_1 := \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2} \cdot \left(0.5 + 0.5 \cdot \cos K\right)}}\\ t_2 := \mathsf{fma}\left(\cos K, 0.5, 0.5\right)\\ t_3 := -2 \cdot \left|J\right|\\ t_4 := \cos \left(\frac{K}{2}\right)\\ t_5 := \left(t\_3 \cdot t\_4\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_4}\right)}^{2}}\\ t_6 := \cos \left(-0.5 \cdot K\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\left(\left(t\_6 \cdot \left|J\right|\right) \cdot \left(\sqrt{\frac{\frac{t\_2 \cdot \left(\left(\left|J\right| \cdot \left|J\right|\right) \cdot 4\right)}{U \cdot U} + 1}{t\_2}} \cdot \left|\frac{U}{t\_0}\right|\right)\right) \cdot -2\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\left(t\_3 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{t\_0 \cdot t\_6}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(U \cdot \mathsf{fma}\left(0.5, \frac{\left|J\right| \cdot t\_6}{{U}^{2} \cdot t\_1}, \left|J\right| \cdot \left(t\_6 \cdot t\_1\right)\right)\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (+ (fabs J) (fabs J)))
       (t_1
        (sqrt
         (/ 0.25 (* (pow (fabs J) 2.0) (+ 0.5 (* 0.5 (cos K)))))))
       (t_2 (fma (cos K) 0.5 0.5))
       (t_3 (* -2.0 (fabs J)))
       (t_4 (cos (/ K 2.0)))
       (t_5
        (*
         (* t_3 t_4)
         (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_4)) 2.0)))))
       (t_6 (cos (* -0.5 K))))
  (*
   (copysign 1.0 J)
   (if (<= t_5 (- INFINITY))
     (*
      (*
       (* t_6 (fabs J))
       (*
        (sqrt
         (/
          (+ (/ (* t_2 (* (* (fabs J) (fabs J)) 4.0)) (* U U)) 1.0)
          t_2))
        (fabs (/ U t_0))))
      -2.0)
     (if (<= t_5 INFINITY)
       (*
        (* t_3 (cos (* K 0.5)))
        (sqrt (+ 1.0 (pow (/ U (* t_0 t_6)) 2.0))))
       (*
        (*
         U
         (fma
          0.5
          (/ (* (fabs J) t_6) (* (pow U 2.0) t_1))
          (* (fabs J) (* t_6 t_1))))
        -2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(J) + fabs(J);
	double t_1 = sqrt((0.25 / (pow(fabs(J), 2.0) * (0.5 + (0.5 * cos(K))))));
	double t_2 = fma(cos(K), 0.5, 0.5);
	double t_3 = -2.0 * fabs(J);
	double t_4 = cos((K / 2.0));
	double t_5 = (t_3 * t_4) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_4)), 2.0)));
	double t_6 = cos((-0.5 * K));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = ((t_6 * fabs(J)) * (sqrt(((((t_2 * ((fabs(J) * fabs(J)) * 4.0)) / (U * U)) + 1.0) / t_2)) * fabs((U / t_0)))) * -2.0;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (t_3 * cos((K * 0.5))) * sqrt((1.0 + pow((U / (t_0 * t_6)), 2.0)));
	} else {
		tmp = (U * fma(0.5, ((fabs(J) * t_6) / (pow(U, 2.0) * t_1)), (fabs(J) * (t_6 * t_1)))) * -2.0;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(abs(J) + abs(J))
	t_1 = sqrt(Float64(0.25 / Float64((abs(J) ^ 2.0) * Float64(0.5 + Float64(0.5 * cos(K))))))
	t_2 = fma(cos(K), 0.5, 0.5)
	t_3 = Float64(-2.0 * abs(J))
	t_4 = cos(Float64(K / 2.0))
	t_5 = Float64(Float64(t_3 * t_4) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_4)) ^ 2.0))))
	t_6 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_6 * abs(J)) * Float64(sqrt(Float64(Float64(Float64(Float64(t_2 * Float64(Float64(abs(J) * abs(J)) * 4.0)) / Float64(U * U)) + 1.0) / t_2)) * abs(Float64(U / t_0)))) * -2.0);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(t_3 * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * t_6)) ^ 2.0))));
	else
		tmp = Float64(Float64(U * fma(0.5, Float64(Float64(abs(J) * t_6) / Float64((U ^ 2.0) * t_1)), Float64(abs(J) * Float64(t_6 * t_1)))) * -2.0);
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.25 / N[(N[Power[N[Abs[J], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 * t$95$4), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[(t$95$6 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(N[(t$95$2 * N[(N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(U * U), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(U / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(t$95$3 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * t$95$6), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(U * N[(0.5 * N[(N[(N[Abs[J], $MachinePrecision] * t$95$6), $MachinePrecision] / N[(N[Power[U, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[J], $MachinePrecision] * N[(t$95$6 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 72.7%

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

   3. Applied rewrites 60.7%

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

   5. Applied rewrites 43.7%

      \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \color{blue}{\left(\sqrt{\frac{\frac{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(\left(J \cdot J\right) \cdot 4\right)}{U \cdot U} + 1}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}} \cdot \left|\frac{U}{J + J}\right|\right)}\right) \cdot -2 \]

   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))))) < +inf.0

   1. Initial program 72.7%

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

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

   3. Applied rewrites 72.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

      15. associate-/r* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 72.7%

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

   7. Applied rewrites 72.7%

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

   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)))))

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

   4. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   6. Applied rewrites 14.3%

      \[\leadsto \color{blue}{\left(U \cdot \mathsf{fma}\left(0.5, \frac{J \cdot \cos \left(-0.5 \cdot K\right)}{{U}^{2} \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot \left(0.5 + 0.5 \cdot \cos K\right)}}}, J \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot \left(0.5 + 0.5 \cdot \cos K\right)}}\right)\right)\right)} \cdot -2 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos K, 0.5, 0.5\right)\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := \left|J\right| + \left|J\right|\\ t_3 := \cos \left(0.5 \cdot K\right)\\ t_4 := \sqrt{\frac{0.25}{{\left(\left|J\right|\right)}^{2} \cdot {t\_3}^{2}}}\\ t_5 := \cos \left(\frac{K}{2}\right)\\ t_6 := \left(t\_1 \cdot t\_5\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_5}\right)}^{2}}\\ t_7 := \cos \left(-0.5 \cdot K\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;\left(\left(t\_7 \cdot \left|J\right|\right) \cdot \left(\sqrt{\frac{\frac{t\_0 \cdot \left(\left(\left|J\right| \cdot \left|J\right|\right) \cdot 4\right)}{U \cdot U} + 1}{t\_0}} \cdot \left|\frac{U}{t\_2}\right|\right)\right) \cdot -2\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\left(t\_1 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{t\_2 \cdot t\_7}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U \cdot \mathsf{fma}\left(-2, \left|J\right| \cdot \left(t\_3 \cdot t\_4\right), -1 \cdot \frac{\left|J\right| \cdot t\_3}{{U}^{2} \cdot t\_4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (fma (cos K) 0.5 0.5))
       (t_1 (* -2.0 (fabs J)))
       (t_2 (+ (fabs J) (fabs J)))
       (t_3 (cos (* 0.5 K)))
       (t_4 (sqrt (/ 0.25 (* (pow (fabs J) 2.0) (pow t_3 2.0)))))
       (t_5 (cos (/ K 2.0)))
       (t_6
        (*
         (* t_1 t_5)
         (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_5)) 2.0)))))
       (t_7 (cos (* -0.5 K))))
  (*
   (copysign 1.0 J)
   (if (<= t_6 (- INFINITY))
     (*
      (*
       (* t_7 (fabs J))
       (*
        (sqrt
         (/
          (+ (/ (* t_0 (* (* (fabs J) (fabs J)) 4.0)) (* U U)) 1.0)
          t_0))
        (fabs (/ U t_2))))
      -2.0)
     (if (<= t_6 INFINITY)
       (*
        (* t_1 (cos (* K 0.5)))
        (sqrt (+ 1.0 (pow (/ U (* t_2 t_7)) 2.0))))
       (*
        U
        (fma
         -2.0
         (* (fabs J) (* t_3 t_4))
         (* -1.0 (/ (* (fabs J) t_3) (* (pow U 2.0) t_4))))))))))

C

double code(double J, double K, double U) {
	double t_0 = fma(cos(K), 0.5, 0.5);
	double t_1 = -2.0 * fabs(J);
	double t_2 = fabs(J) + fabs(J);
	double t_3 = cos((0.5 * K));
	double t_4 = sqrt((0.25 / (pow(fabs(J), 2.0) * pow(t_3, 2.0))));
	double t_5 = cos((K / 2.0));
	double t_6 = (t_1 * t_5) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_5)), 2.0)));
	double t_7 = cos((-0.5 * K));
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = ((t_7 * fabs(J)) * (sqrt(((((t_0 * ((fabs(J) * fabs(J)) * 4.0)) / (U * U)) + 1.0) / t_0)) * fabs((U / t_2)))) * -2.0;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = (t_1 * cos((K * 0.5))) * sqrt((1.0 + pow((U / (t_2 * t_7)), 2.0)));
	} else {
		tmp = U * fma(-2.0, (fabs(J) * (t_3 * t_4)), (-1.0 * ((fabs(J) * t_3) / (pow(U, 2.0) * t_4))));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = fma(cos(K), 0.5, 0.5)
	t_1 = Float64(-2.0 * abs(J))
	t_2 = Float64(abs(J) + abs(J))
	t_3 = cos(Float64(0.5 * K))
	t_4 = sqrt(Float64(0.25 / Float64((abs(J) ^ 2.0) * (t_3 ^ 2.0))))
	t_5 = cos(Float64(K / 2.0))
	t_6 = Float64(Float64(t_1 * t_5) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_5)) ^ 2.0))))
	t_7 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_7 * abs(J)) * Float64(sqrt(Float64(Float64(Float64(Float64(t_0 * Float64(Float64(abs(J) * abs(J)) * 4.0)) / Float64(U * U)) + 1.0) / t_0)) * abs(Float64(U / t_2)))) * -2.0);
	elseif (t_6 <= Inf)
		tmp = Float64(Float64(t_1 * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + (Float64(U / Float64(t_2 * t_7)) ^ 2.0))));
	else
		tmp = Float64(U * fma(-2.0, Float64(abs(J) * Float64(t_3 * t_4)), Float64(-1.0 * Float64(Float64(abs(J) * t_3) / Float64((U ^ 2.0) * t_4)))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(0.25 / N[(N[Power[N[Abs[J], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$1 * t$95$5), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$6, (-Infinity)], N[(N[(N[(t$95$7 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(N[(t$95$0 * N[(N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(U * U), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(U / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(N[(t$95$1 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$2 * t$95$7), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(U * N[(-2.0 * N[(N[Abs[J], $MachinePrecision] * N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[(N[Abs[J], $MachinePrecision] * t$95$3), $MachinePrecision] / N[(N[Power[U, 2.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

   5. Applied rewrites 43.7%

      \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \color{blue}{\left(\sqrt{\frac{\frac{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(\left(J \cdot J\right) \cdot 4\right)}{U \cdot U} + 1}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}} \cdot \left|\frac{U}{J + J}\right|\right)}\right) \cdot -2 \]

   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))))) < +inf.0

   1. Initial program 72.7%

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

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

   3. Applied rewrites 72.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

      15. associate-/r* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 72.7%

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

   7. Applied rewrites 72.7%

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

   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)))))

   1. Initial program 72.7%

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

   2. Taylor expanded in J around inf

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in K around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 27.8%

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

   7. Applied rewrites 27.8%

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

   8. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   10. Applied rewrites 14.3%

       \[\leadsto \color{blue}{U \cdot \mathsf{fma}\left(-2, J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right), -1 \cdot \frac{J \cdot \cos \left(0.5 \cdot K\right)}{{U}^{2} \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := -2 \cdot \left|J\right|\\ t_3 := \left(t\_2 \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ t_4 := \left(t\_0 \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right) \cdot -2\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\left(t\_2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(\left|J\right| + \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -0.5 K)))
       (t_1 (cos (/ K 2.0)))
       (t_2 (* -2.0 (fabs J)))
       (t_3
        (*
         (* t_2 t_1)
         (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_1)) 2.0)))))
       (t_4
        (*
         (*
          t_0
          (sqrt (* 0.25 (/ (pow U 2.0) (+ 0.5 (* 0.5 (cos K)))))))
         -2.0)))
  (*
   (copysign 1.0 J)
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 INFINITY)
       (*
        (* t_2 (cos (* K 0.5)))
        (sqrt (+ 1.0 (pow (/ U (* (+ (fabs J) (fabs J)) t_0)) 2.0))))
       t_4)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = -2.0 * fabs(J);
	double t_3 = (t_2 * t_1) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double t_4 = (t_0 * sqrt((0.25 * (pow(U, 2.0) / (0.5 + (0.5 * cos(K))))))) * -2.0;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_2 * cos((K * 0.5))) * sqrt((1.0 + pow((U / ((fabs(J) + fabs(J)) * t_0)), 2.0)));
	} else {
		tmp = t_4;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((-0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = -2.0 * Math.abs(J);
	double t_3 = (t_2 * t_1) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double t_4 = (t_0 * Math.sqrt((0.25 * (Math.pow(U, 2.0) / (0.5 + (0.5 * Math.cos(K))))))) * -2.0;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_4;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (t_2 * Math.cos((K * 0.5))) * Math.sqrt((1.0 + Math.pow((U / ((Math.abs(J) + Math.abs(J)) * t_0)), 2.0)));
	} else {
		tmp = t_4;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = -2.0 * math.fabs(J)
	t_3 = (t_2 * t_1) * math.sqrt((1.0 + math.pow((U / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	t_4 = (t_0 * math.sqrt((0.25 * (math.pow(U, 2.0) / (0.5 + (0.5 * math.cos(K))))))) * -2.0
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_4
	elif t_3 <= math.inf:
		tmp = (t_2 * math.cos((K * 0.5))) * math.sqrt((1.0 + math.pow((U / ((math.fabs(J) + math.fabs(J)) * t_0)), 2.0)))
	else:
		tmp = t_4
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(-2.0 * abs(J))
	t_3 = Float64(Float64(t_2 * t_1) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	t_4 = Float64(Float64(t_0 * sqrt(Float64(0.25 * Float64((U ^ 2.0) / Float64(0.5 + Float64(0.5 * cos(K))))))) * -2.0)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_2 * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(abs(J) + abs(J)) * t_0)) ^ 2.0))));
	else
		tmp = t_4;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((-0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = -2.0 * abs(J);
	t_3 = (t_2 * t_1) * sqrt((1.0 + ((U / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	t_4 = (t_0 * sqrt((0.25 * ((U ^ 2.0) / (0.5 + (0.5 * cos(K))))))) * -2.0;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = (t_2 * cos((K * 0.5))) * sqrt((1.0 + ((U / ((abs(J) + abs(J)) * t_0)) ^ 2.0)));
	else
		tmp = t_4;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := 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[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 * N[Sqrt[N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, Infinity], N[(N[(t$95$2 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$4]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

   4. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 15.2%

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

   6. Applied rewrites 15.2%

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

   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))))) < +inf.0

   1. Initial program 72.7%

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

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

   3. Applied rewrites 72.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 72.7%

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

   5. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lift-cos.f64 N/A

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

      15. associate-/r* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 72.7%

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

   7. Applied rewrites 72.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + \color{blue}{{\left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := -2 \cdot \left|J\right|\\ t_3 := \left(t\_2 \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ t_4 := \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right) \cdot -2\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\left(\sqrt{{\left(\frac{U}{\left(\left|J\right| + \left|J\right|\right) \cdot t\_0}\right)}^{2} - -1} \cdot t\_0\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* 0.5 K)))
       (t_1 (cos (/ K 2.0)))
       (t_2 (* -2.0 (fabs J)))
       (t_3
        (*
         (* t_2 t_1)
         (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_1)) 2.0)))))
       (t_4
        (*
         (*
          (cos (* -0.5 K))
          (sqrt (* 0.25 (/ (pow U 2.0) (+ 0.5 (* 0.5 (cos K)))))))
         -2.0)))
  (*
   (copysign 1.0 J)
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 INFINITY)
       (*
        (*
         (sqrt (- (pow (/ U (* (+ (fabs J) (fabs J)) t_0)) 2.0) -1.0))
         t_0)
        t_2)
       t_4)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = -2.0 * fabs(J);
	double t_3 = (t_2 * t_1) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double t_4 = (cos((-0.5 * K)) * sqrt((0.25 * (pow(U, 2.0) / (0.5 + (0.5 * cos(K))))))) * -2.0;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((pow((U / ((fabs(J) + fabs(J)) * t_0)), 2.0) - -1.0)) * t_0) * t_2;
	} else {
		tmp = t_4;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = -2.0 * Math.abs(J);
	double t_3 = (t_2 * t_1) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double t_4 = (Math.cos((-0.5 * K)) * Math.sqrt((0.25 * (Math.pow(U, 2.0) / (0.5 + (0.5 * Math.cos(K))))))) * -2.0;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_4;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.sqrt((Math.pow((U / ((Math.abs(J) + Math.abs(J)) * t_0)), 2.0) - -1.0)) * t_0) * t_2;
	} else {
		tmp = t_4;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((0.5 * K))
	t_1 = math.cos((K / 2.0))
	t_2 = -2.0 * math.fabs(J)
	t_3 = (t_2 * t_1) * math.sqrt((1.0 + math.pow((U / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	t_4 = (math.cos((-0.5 * K)) * math.sqrt((0.25 * (math.pow(U, 2.0) / (0.5 + (0.5 * math.cos(K))))))) * -2.0
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_4
	elif t_3 <= math.inf:
		tmp = (math.sqrt((math.pow((U / ((math.fabs(J) + math.fabs(J)) * t_0)), 2.0) - -1.0)) * t_0) * t_2
	else:
		tmp = t_4
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(-2.0 * abs(J))
	t_3 = Float64(Float64(t_2 * t_1) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	t_4 = Float64(Float64(cos(Float64(-0.5 * K)) * sqrt(Float64(0.25 * Float64((U ^ 2.0) / Float64(0.5 + Float64(0.5 * cos(K))))))) * -2.0)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64((Float64(U / Float64(Float64(abs(J) + abs(J)) * t_0)) ^ 2.0) - -1.0)) * t_0) * t_2);
	else
		tmp = t_4;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((0.5 * K));
	t_1 = cos((K / 2.0));
	t_2 = -2.0 * abs(J);
	t_3 = (t_2 * t_1) * sqrt((1.0 + ((U / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	t_4 = (cos((-0.5 * K)) * sqrt((0.25 * ((U ^ 2.0) / (0.5 + (0.5 * cos(K))))))) * -2.0;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = (sqrt((((U / ((abs(J) + abs(J)) * t_0)) ^ 2.0) - -1.0)) * t_0) * t_2;
	else
		tmp = t_4;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := 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[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(N[Power[N[(U / N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision], t$95$4]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

   4. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 15.2%

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

   6. Applied rewrites 15.2%

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

   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))))) < +inf.0

   1. Initial program 72.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 85.1%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 85.1%

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.1%

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

   3. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.2%

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. cos-neg-rev N/A

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

      17. lift-cos.f64 85.2%

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

   5. Applied rewrites 85.2%

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

   7. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\left(\sqrt{{\left(\frac{U}{\left(J + J\right) \cdot \cos \left(0.5 \cdot K\right)}\right)}^{2} - -1} \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(-2 \cdot J\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-0.5 \cdot K\right)\\ \left(\left(t\_0 \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot t\_0}\right) \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (cos (* -0.5 K))))
  (* (* (* t_0 J) -2.0) (cosh (asinh (/ U (* (+ J J) t_0)))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((-0.5 * K));
	return ((t_0 * J) * -2.0) * cosh(asinh((U / ((J + J) * t_0))));
}

Python

def code(J, K, U):
	t_0 = math.cos((-0.5 * K))
	return ((t_0 * J) * -2.0) * math.cosh(math.asinh((U / ((J + J) * t_0))))

Julia

function code(J, K, U)
	t_0 = cos(Float64(-0.5 * K))
	return Float64(Float64(Float64(t_0 * J) * -2.0) * cosh(asinh(Float64(U / Float64(Float64(J + J) * t_0)))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((-0.5 * K));
	tmp = ((t_0 * J) * -2.0) * cosh(asinh((U / ((J + J) * t_0))));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(t$95$0 * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U / N[(N[(J + J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.7%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. cosh-asinh-rev N/A

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

   7. lower-cosh.f64 N/A

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

   8. lower-asinh.f64 85.1%

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

   9. lift-*.f64 N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f64 85.1%

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

   12. lift-cos.f64 N/A

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

   13. cos-neg-rev N/A

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

   14. lower-cos.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. distribute-neg-frac2 N/A

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

   17. metadata-eval N/A

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

   18. mult-flip-rev N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. metadata-eval 85.1%

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

3. Applied rewrites 85.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 85.2%

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

   8. lift-cos.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. mult-flip N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. cos-neg-rev N/A

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

   17. lift-cos.f64 85.2%

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

5. Applied rewrites 85.2%

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

Alternative 6: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{U}{\left|J\right|}\\ t_1 := 0.5 + 0.5 \cdot \cos K\\ t_2 := \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{t\_1}}\right) \cdot -2\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(-2 \cdot \left|J\right|\right) \cdot t\_3\\ t_5 := t\_4 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_3}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_4 \cdot \sqrt{1 + \frac{\frac{t\_0 \cdot t\_0}{4}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (/ U (fabs J)))
       (t_1 (+ 0.5 (* 0.5 (cos K))))
       (t_2
        (*
         (* (cos (* -0.5 K)) (sqrt (* 0.25 (/ (pow U 2.0) t_1))))
         -2.0))
       (t_3 (cos (/ K 2.0)))
       (t_4 (* (* -2.0 (fabs J)) t_3))
       (t_5
        (*
         t_4
         (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_3)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_5 (- INFINITY))
     t_2
     (if (<= t_5 INFINITY)
       (* t_4 (sqrt (+ 1.0 (/ (/ (* t_0 t_0) 4.0) t_1))))
       t_2)))))

C

double code(double J, double K, double U) {
	double t_0 = U / fabs(J);
	double t_1 = 0.5 + (0.5 * cos(K));
	double t_2 = (cos((-0.5 * K)) * sqrt((0.25 * (pow(U, 2.0) / t_1)))) * -2.0;
	double t_3 = cos((K / 2.0));
	double t_4 = (-2.0 * fabs(J)) * t_3;
	double t_5 = t_4 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_3)), 2.0)));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_4 * sqrt((1.0 + (((t_0 * t_0) / 4.0) / t_1)));
	} else {
		tmp = t_2;
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = U / Math.abs(J);
	double t_1 = 0.5 + (0.5 * Math.cos(K));
	double t_2 = (Math.cos((-0.5 * K)) * Math.sqrt((0.25 * (Math.pow(U, 2.0) / t_1)))) * -2.0;
	double t_3 = Math.cos((K / 2.0));
	double t_4 = (-2.0 * Math.abs(J)) * t_3;
	double t_5 = t_4 * Math.sqrt((1.0 + Math.pow((U / ((2.0 * Math.abs(J)) * t_3)), 2.0)));
	double tmp;
	if (t_5 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = t_4 * Math.sqrt((1.0 + (((t_0 * t_0) / 4.0) / t_1)));
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = U / math.fabs(J)
	t_1 = 0.5 + (0.5 * math.cos(K))
	t_2 = (math.cos((-0.5 * K)) * math.sqrt((0.25 * (math.pow(U, 2.0) / t_1)))) * -2.0
	t_3 = math.cos((K / 2.0))
	t_4 = (-2.0 * math.fabs(J)) * t_3
	t_5 = t_4 * math.sqrt((1.0 + math.pow((U / ((2.0 * math.fabs(J)) * t_3)), 2.0)))
	tmp = 0
	if t_5 <= -math.inf:
		tmp = t_2
	elif t_5 <= math.inf:
		tmp = t_4 * math.sqrt((1.0 + (((t_0 * t_0) / 4.0) / t_1)))
	else:
		tmp = t_2
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(U / abs(J))
	t_1 = Float64(0.5 + Float64(0.5 * cos(K)))
	t_2 = Float64(Float64(cos(Float64(-0.5 * K)) * sqrt(Float64(0.25 * Float64((U ^ 2.0) / t_1)))) * -2.0)
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(-2.0 * abs(J)) * t_3)
	t_5 = Float64(t_4 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_3)) ^ 2.0))))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_5 <= Inf)
		tmp = Float64(t_4 * sqrt(Float64(1.0 + Float64(Float64(Float64(t_0 * t_0) / 4.0) / t_1))));
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = U / abs(J);
	t_1 = 0.5 + (0.5 * cos(K));
	t_2 = (cos((-0.5 * K)) * sqrt((0.25 * ((U ^ 2.0) / t_1)))) * -2.0;
	t_3 = cos((K / 2.0));
	t_4 = (-2.0 * abs(J)) * t_3;
	t_5 = t_4 * sqrt((1.0 + ((U / ((2.0 * abs(J)) * t_3)) ^ 2.0)));
	tmp = 0.0;
	if (t_5 <= -Inf)
		tmp = t_2;
	elseif (t_5 <= Inf)
		tmp = t_4 * sqrt((1.0 + (((t_0 * t_0) / 4.0) / t_1)));
	else
		tmp = t_2;
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, (-Infinity)], t$95$2, If[LessEqual[t$95$5, Infinity], N[(t$95$4 * N[Sqrt[N[(1.0 + N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

   4. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 15.2%

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

   6. Applied rewrites 15.2%

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

   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))))) < +inf.0

   1. Initial program 72.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 72.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|J\right| + \left|J\right|\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \left(t\_1 \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right) \cdot -2\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_3\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_3}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\left(\left(t\_1 \cdot \left|J\right|\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{t\_0}, \frac{U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot t\_0}, 1\right)}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (+ (fabs J) (fabs J)))
       (t_1 (cos (* -0.5 K)))
       (t_2
        (*
         (*
          t_1
          (sqrt (* 0.25 (/ (pow U 2.0) (+ 0.5 (* 0.5 (cos K)))))))
         -2.0))
       (t_3 (cos (/ K 2.0)))
       (t_4
        (*
         (* (* -2.0 (fabs J)) t_3)
         (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_3)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_4 (- INFINITY))
     t_2
     (if (<= t_4 INFINITY)
       (*
        (*
         (* t_1 (fabs J))
         (sqrt
          (fma (/ U t_0) (/ U (* (fma (cos K) 0.5 0.5) t_0)) 1.0)))
        -2.0)
       t_2)))))

C

double code(double J, double K, double U) {
	double t_0 = fabs(J) + fabs(J);
	double t_1 = cos((-0.5 * K));
	double t_2 = (t_1 * sqrt((0.25 * (pow(U, 2.0) / (0.5 + (0.5 * cos(K))))))) * -2.0;
	double t_3 = cos((K / 2.0));
	double t_4 = ((-2.0 * fabs(J)) * t_3) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_3)), 2.0)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = ((t_1 * fabs(J)) * sqrt(fma((U / t_0), (U / (fma(cos(K), 0.5, 0.5) * t_0)), 1.0))) * -2.0;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(abs(J) + abs(J))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = Float64(Float64(t_1 * sqrt(Float64(0.25 * Float64((U ^ 2.0) / Float64(0.5 + Float64(0.5 * cos(K))))))) * -2.0)
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(Float64(-2.0 * abs(J)) * t_3) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_3)) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(t_1 * abs(J)) * sqrt(fma(Float64(U / t_0), Float64(U / Float64(fma(cos(K), 0.5, 0.5) * t_0)), 1.0))) * -2.0);
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Sqrt[N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], t$95$2, If[LessEqual[t$95$4, Infinity], N[(N[(N[(t$95$1 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U / t$95$0), $MachinePrecision] * N[(U / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$2]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

   4. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 15.2%

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

   6. Applied rewrites 15.2%

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

   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))))) < +inf.0

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 72.6%

      \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(J + J\right)}, 1\right)}}\right) \cdot -2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := 1 + -0.125 \cdot {K}^{2}\\ t_1 := \left|J\right| + \left|J\right|\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(-2 \cdot \left|J\right|\right) \cdot t\_2\\ t_4 := t\_3 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_3 \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\left|J\right|}\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot \left|J\right|\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{t\_1}, \frac{U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot t\_1}, 1\right)}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left|J\right|\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{t\_1 \cdot t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (+ 1.0 (* -0.125 (pow K 2.0))))
       (t_1 (+ (fabs J) (fabs J)))
       (t_2 (cos (/ K 2.0)))
       (t_3 (* (* -2.0 (fabs J)) t_2))
       (t_4
        (*
         t_3
         (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_2)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_4 (- INFINITY))
     (* t_3 (cosh (asinh (* 0.5 (/ U (fabs J))))))
     (if (<= t_4 INFINITY)
       (*
        (*
         (* (cos (* -0.5 K)) (fabs J))
         (sqrt
          (fma (/ U t_1) (/ U (* (fma (cos K) 0.5 0.5) t_1)) 1.0)))
        -2.0)
       (*
        (* (* t_0 (fabs J)) -2.0)
        (cosh (asinh (/ U (* t_1 t_0))))))))))

C

double code(double J, double K, double U) {
	double t_0 = 1.0 + (-0.125 * pow(K, 2.0));
	double t_1 = fabs(J) + fabs(J);
	double t_2 = cos((K / 2.0));
	double t_3 = (-2.0 * fabs(J)) * t_2;
	double t_4 = t_3 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_3 * cosh(asinh((0.5 * (U / fabs(J)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = ((cos((-0.5 * K)) * fabs(J)) * sqrt(fma((U / t_1), (U / (fma(cos(K), 0.5, 0.5) * t_1)), 1.0))) * -2.0;
	} else {
		tmp = ((t_0 * fabs(J)) * -2.0) * cosh(asinh((U / (t_1 * t_0))));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(1.0 + Float64(-0.125 * (K ^ 2.0)))
	t_1 = Float64(abs(J) + abs(J))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(-2.0 * abs(J)) * t_2)
	t_4 = Float64(t_3 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(t_3 * cosh(asinh(Float64(0.5 * Float64(U / abs(J))))));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * abs(J)) * sqrt(fma(Float64(U / t_1), Float64(U / Float64(fma(cos(K), 0.5, 0.5) * t_1)), 1.0))) * -2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * abs(J)) * -2.0) * cosh(asinh(Float64(U / Float64(t_1 * t_0)))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], N[(t$95$3 * N[Cosh[N[ArcSinh[N[(0.5 * N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U / t$95$1), $MachinePrecision] * N[(U / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 85.1%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 85.1%

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.1%

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.7%

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

   6. Applied rewrites 71.7%

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

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

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 72.6%

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

   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)))))

   1. Initial program 72.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 85.1%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 85.1%

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.1%

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

   3. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.2%

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. cos-neg-rev N/A

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

      17. lift-cos.f64 85.2%

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\frac{-1}{8} \cdot {K}^{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot \color{blue}{{K}^{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \]

      3. lower-pow.f64 44.8%

         \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]

   8. Applied rewrites 44.8%

      \[\leadsto \left(\left(\color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]

   9. Taylor expanded in K around 0

      \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + \color{blue}{\frac{-1}{8} \cdot {K}^{2}}\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + \frac{-1}{8} \cdot \color{blue}{{K}^{2}}\right)}\right) \]

       3. lower-pow.f64 47.6%

          \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right)}\right) \]

   11. Applied rewrites 47.6%

       \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := 1 + -0.125 \cdot {K}^{2}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot \left|J\right|\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 10^{-193}:\\ \;\;\;\;t\_2 \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\left|J\right|}\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+292}:\\ \;\;\;\;\left(\left(\sqrt{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(\left(\left|J\right| \cdot \left|J\right|\right) \cdot 4\right)}, U, 1\right)} \cdot \left|J\right|\right) \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left|J\right|\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(\left|J\right| + \left|J\right|\right) \cdot t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (+ 1.0 (* -0.125 (pow K 2.0))))
       (t_1 (cos (/ K 2.0)))
       (t_2 (* (* -2.0 (fabs J)) t_1))
       (t_3
        (*
         t_2
         (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_1)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_3 1e-193)
     (* t_2 (cosh (asinh (* 0.5 (/ U (fabs J))))))
     (if (<= t_3 1e+292)
       (*
        (*
         (*
          (sqrt
           (fma
            (/
             U
             (* (fma (cos K) 0.5 0.5) (* (* (fabs J) (fabs J)) 4.0)))
            U
            1.0))
          (fabs J))
         (cos (* -0.5 K)))
        -2.0)
       (*
        (* (* t_0 (fabs J)) -2.0)
        (cosh (asinh (/ U (* (+ (fabs J) (fabs J)) t_0))))))))))

C

double code(double J, double K, double U) {
	double t_0 = 1.0 + (-0.125 * pow(K, 2.0));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * fabs(J)) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= 1e-193) {
		tmp = t_2 * cosh(asinh((0.5 * (U / fabs(J)))));
	} else if (t_3 <= 1e+292) {
		tmp = ((sqrt(fma((U / (fma(cos(K), 0.5, 0.5) * ((fabs(J) * fabs(J)) * 4.0))), U, 1.0)) * fabs(J)) * cos((-0.5 * K))) * -2.0;
	} else {
		tmp = ((t_0 * fabs(J)) * -2.0) * cosh(asinh((U / ((fabs(J) + fabs(J)) * t_0))));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(1.0 + Float64(-0.125 * (K ^ 2.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * abs(J)) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 1e-193)
		tmp = Float64(t_2 * cosh(asinh(Float64(0.5 * Float64(U / abs(J))))));
	elseif (t_3 <= 1e+292)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(U / Float64(fma(cos(K), 0.5, 0.5) * Float64(Float64(abs(J) * abs(J)) * 4.0))), U, 1.0)) * abs(J)) * cos(Float64(-0.5 * K))) * -2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * abs(J)) * -2.0) * cosh(asinh(Float64(U / Float64(Float64(abs(J) + abs(J)) * t_0)))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, 1e-193], N[(t$95$2 * N[Cosh[N[ArcSinh[N[(0.5 * N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+292], N[(N[(N[(N[Sqrt[N[(N[(U / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U + 1.0), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U / N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1e-193

   1. Initial program 72.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 85.1%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 85.1%

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.1%

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.7%

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

   6. Applied rewrites 71.7%

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

   if 1e-193 < (*.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))))) < 1e292

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. mult-flip N/A

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

      13. lift-/.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(\left(J \cdot J\right) \cdot 4\right)}, U, 1\right)} \cdot J\right) \cdot \cos \left(-0.5 \cdot K\right)\right)} \cdot -2 \]

   if 1e292 < (*.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 72.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 85.1%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 85.1%

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.1%

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

   3. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.2%

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. cos-neg-rev N/A

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

      17. lift-cos.f64 85.2%

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\frac{-1}{8} \cdot {K}^{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot \color{blue}{{K}^{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \]

      3. lower-pow.f64 44.8%

         \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]

   8. Applied rewrites 44.8%

      \[\leadsto \left(\left(\color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]

   9. Taylor expanded in K around 0

      \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + \color{blue}{\frac{-1}{8} \cdot {K}^{2}}\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + \frac{-1}{8} \cdot \color{blue}{{K}^{2}}\right)}\right) \]

       3. lower-pow.f64 47.6%

          \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right)}\right) \]

   11. Applied rewrites 47.6%

       \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := 1 + -0.125 \cdot {K}^{2}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot \left|J\right|\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 10^{-193}:\\ \;\;\;\;t\_2 \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\left|J\right|}\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+292}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot \left|J\right|\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(\left(\left|J\right| \cdot \left|J\right|\right) \cdot 4\right)}, U, 1\right)}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left|J\right|\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(\left|J\right| + \left|J\right|\right) \cdot t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (let* ((t_0 (+ 1.0 (* -0.125 (pow K 2.0))))
       (t_1 (cos (/ K 2.0)))
       (t_2 (* (* -2.0 (fabs J)) t_1))
       (t_3
        (*
         t_2
         (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_1)) 2.0))))))
  (*
   (copysign 1.0 J)
   (if (<= t_3 1e-193)
     (* t_2 (cosh (asinh (* 0.5 (/ U (fabs J))))))
     (if (<= t_3 1e+292)
       (*
        (*
         (* (cos (* -0.5 K)) (fabs J))
         (sqrt
          (fma
           (/
            U
            (* (fma (cos K) 0.5 0.5) (* (* (fabs J) (fabs J)) 4.0)))
           U
           1.0)))
        -2.0)
       (*
        (* (* t_0 (fabs J)) -2.0)
        (cosh (asinh (/ U (* (+ (fabs J) (fabs J)) t_0))))))))))

C

double code(double J, double K, double U) {
	double t_0 = 1.0 + (-0.125 * pow(K, 2.0));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * fabs(J)) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= 1e-193) {
		tmp = t_2 * cosh(asinh((0.5 * (U / fabs(J)))));
	} else if (t_3 <= 1e+292) {
		tmp = ((cos((-0.5 * K)) * fabs(J)) * sqrt(fma((U / (fma(cos(K), 0.5, 0.5) * ((fabs(J) * fabs(J)) * 4.0))), U, 1.0))) * -2.0;
	} else {
		tmp = ((t_0 * fabs(J)) * -2.0) * cosh(asinh((U / ((fabs(J) + fabs(J)) * t_0))));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(1.0 + Float64(-0.125 * (K ^ 2.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * abs(J)) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 1e-193)
		tmp = Float64(t_2 * cosh(asinh(Float64(0.5 * Float64(U / abs(J))))));
	elseif (t_3 <= 1e+292)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * abs(J)) * sqrt(fma(Float64(U / Float64(fma(cos(K), 0.5, 0.5) * Float64(Float64(abs(J) * abs(J)) * 4.0))), U, 1.0))) * -2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * abs(J)) * -2.0) * cosh(asinh(Float64(U / Float64(Float64(abs(J) + abs(J)) * t_0)))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, 1e-193], N[(t$95$2 * N[Cosh[N[ArcSinh[N[(0.5 * N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+292], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(N[(N[Abs[J], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U / N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1e-193

   1. Initial program 72.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 85.1%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 85.1%

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.1%

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 71.7%

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

   6. Applied rewrites 71.7%

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

   if 1e-193 < (*.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))))) < 1e292

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.7%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 60.7%

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. count-2-rev N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. count-2-rev N/A

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

      15. *-commutative N/A

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

      16. swap-sqr N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 60.7%

          \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(\left(J \cdot J\right) \cdot \color{blue}{4}\right)}, U, 1\right)}\right) \cdot -2 \]

   5. Applied rewrites 60.7%

      \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\color{blue}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(\left(J \cdot J\right) \cdot 4\right)}}, U, 1\right)}\right) \cdot -2 \]

   if 1e292 < (*.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 72.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. cosh-asinh-rev N/A

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

      7. lower-cosh.f64 N/A

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

      8. lower-asinh.f64 85.1%

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 85.1%

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

      12. lift-cos.f64 N/A

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

      13. cos-neg-rev N/A

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

      14. lower-cos.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-neg-frac2 N/A

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

      17. metadata-eval N/A

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

      18. mult-flip-rev N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.1%

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

   3. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.2%

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. cos-neg-rev N/A

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

      17. lift-cos.f64 85.2%

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(1 + \color{blue}{\frac{-1}{8} \cdot {K}^{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot \color{blue}{{K}^{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \]

      3. lower-pow.f64 44.8%

         \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]

   8. Applied rewrites 44.8%

      \[\leadsto \left(\left(\color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right) \]

   9. Taylor expanded in K around 0

      \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + \color{blue}{\frac{-1}{8} \cdot {K}^{2}}\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + \frac{-1}{8} \cdot \color{blue}{{K}^{2}}\right)}\right) \]

       3. lower-pow.f64 47.6%

          \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \left(1 + -0.125 \cdot {K}^{\color{blue}{2}}\right)}\right) \]

   11. Applied rewrites 47.6%

       \[\leadsto \left(\left(\left(1 + -0.125 \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 71.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (cosh (asinh (* 0.5 (/ U J))))))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * cos((K / 2.0))) * cosh(asinh((0.5 * (U / J))));
}

Python

def code(J, K, U):
	return ((-2.0 * J) * math.cos((K / 2.0))) * math.cosh(math.asinh((0.5 * (U / J))))

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * cosh(asinh(Float64(0.5 * Float64(U / J)))))
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * cos((K / 2.0))) * cosh(asinh((0.5 * (U / J))));
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. cosh-asinh-rev N/A

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

   7. lower-cosh.f64 N/A

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

   8. lower-asinh.f64 85.1%

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

   9. lift-*.f64 N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f64 85.1%

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

   12. lift-cos.f64 N/A

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

   13. cos-neg-rev N/A

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

   14. lower-cos.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. distribute-neg-frac2 N/A

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

   17. metadata-eval N/A

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

   18. mult-flip-rev N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. metadata-eval 85.1%

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

3. Applied rewrites 85.1%

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

4. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 71.7%

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

6. Applied rewrites 71.7%

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

Alternative 12: 64.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (*
 (* (* -2.0 J) (cos (/ K 2.0)))
 (sqrt (+ 1.0 (pow (* 0.5 (/ U J)) 2.0)))))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + pow((0.5 * (U / J)), 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
    code = (((-2.0d0) * j) * cos((k / 2.0d0))) * sqrt((1.0d0 + ((0.5d0 * (u / j)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	return ((-2.0 * J) * Math.cos((K / 2.0))) * Math.sqrt((1.0 + Math.pow((0.5 * (U / J)), 2.0)));
}

Python

def code(J, K, U):
	return ((-2.0 * J) * math.cos((K / 2.0))) * math.sqrt((1.0 + math.pow((0.5 * (U / J)), 2.0)))

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U / J)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + ((0.5 * (U / J)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

2. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.0%

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

4. Applied rewrites 64.0%

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

Alternative 13: 62.2% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (*
 (* (* -2.0 J) (cos (/ K 2.0)))
 (sqrt (fma (* (/ U (+ J J)) U) (/ 0.5 J) 1.0))))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * cos((K / 2.0))) * sqrt(fma(((U / (J + J)) * U), (0.5 / J), 1.0));
}

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * sqrt(fma(Float64(Float64(U / Float64(J + J)) * U), Float64(0.5 / J), 1.0)))
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(U / N[(J + J), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * N[(0.5 / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f64 N/A

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

   9. frac-times N/A

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

   10. mult-flip N/A

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

   11. lower-fma.f64 N/A

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

3. Applied rewrites 69.8%

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

4. Taylor expanded in K around 0

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

   1. lower-/.f64 62.2%

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

6. Applied rewrites 62.2%

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

Alternative 14: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|K\right| \leq 132000000000:\\ \;\;\;\;-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(-0.5 \cdot \left|K\right|\right)\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
  :precision binary64
  (if (<= (fabs K) 132000000000.0)
  (* -2.0 (* J (sqrt (+ 1.0 (* 0.25 (/ (pow U 2.0) (pow J 2.0)))))))
  (* -2.0 (* J (cos (* -0.5 (fabs K)))))))

C

double code(double J, double K, double U) {
	double tmp;
	if (fabs(K) <= 132000000000.0) {
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * (pow(U, 2.0) / pow(J, 2.0))))));
	} else {
		tmp = -2.0 * (J * cos((-0.5 * fabs(K))));
	}
	return tmp;
}

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) :: tmp
    if (abs(k) <= 132000000000.0d0) then
        tmp = (-2.0d0) * (j * sqrt((1.0d0 + (0.25d0 * ((u ** 2.0d0) / (j ** 2.0d0))))))
    else
        tmp = (-2.0d0) * (j * cos(((-0.5d0) * abs(k))))
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (Math.abs(K) <= 132000000000.0) {
		tmp = -2.0 * (J * Math.sqrt((1.0 + (0.25 * (Math.pow(U, 2.0) / Math.pow(J, 2.0))))));
	} else {
		tmp = -2.0 * (J * Math.cos((-0.5 * Math.abs(K))));
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if math.fabs(K) <= 132000000000.0:
		tmp = -2.0 * (J * math.sqrt((1.0 + (0.25 * (math.pow(U, 2.0) / math.pow(J, 2.0))))))
	else:
		tmp = -2.0 * (J * math.cos((-0.5 * math.fabs(K))))
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (abs(K) <= 132000000000.0)
		tmp = Float64(-2.0 * Float64(J * sqrt(Float64(1.0 + Float64(0.25 * Float64((U ^ 2.0) / (J ^ 2.0)))))));
	else
		tmp = Float64(-2.0 * Float64(J * cos(Float64(-0.5 * abs(K)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (abs(K) <= 132000000000.0)
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * ((U ^ 2.0) / (J ^ 2.0))))));
	else
		tmp = -2.0 * (J * cos((-0.5 * abs(K))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[N[Abs[K], $MachinePrecision], 132000000000.0], N[(-2.0 * N[(J * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / N[Power[J, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(J * N[Cos[N[(-0.5 * N[Abs[K], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.32e11

   1. Initial program 72.7%

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

   2. 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. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 32.5%

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

   4. Applied rewrites 32.5%

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

   if 1.32e11 < K

   1. Initial program 72.7%

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

   3. Applied rewrites 60.7%

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

   4. Taylor expanded in J around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 51.6%

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

   6. Applied rewrites 51.6%

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(-0.5 \cdot K\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.6% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (* -2.0 (* J (cos (* -0.5 K)))))

C

double code(double J, double K, double U) {
	return -2.0 * (J * cos((-0.5 * K)));
}

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
    code = (-2.0d0) * (j * cos(((-0.5d0) * k)))
end function

Java

public static double code(double J, double K, double U) {
	return -2.0 * (J * Math.cos((-0.5 * K)));
}

Python

def code(J, K, U):
	return -2.0 * (J * math.cos((-0.5 * K)))

Julia

function code(J, K, U)
	return Float64(-2.0 * Float64(J * cos(Float64(-0.5 * K))))
end

MATLAB

function tmp = code(J, K, U)
	tmp = -2.0 * (J * cos((-0.5 * K)));
end

Wolfram

code[J_, K_, U_] := N[(-2.0 * N[(J * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

3. Applied rewrites 60.7%

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

4. Taylor expanded in J around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 51.6%

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

6. Applied rewrites 51.6%

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

Alternative 16: 27.8% accurate, 5.8× speedup

Math

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

FPCore

(FPCore (J K U)
  :precision binary64
  (* (fma (* (* 0.25 J) K) K (* -2.0 J)) (sqrt 1.0)))

C

double code(double J, double K, double U) {
	return fma(((0.25 * J) * K), K, (-2.0 * J)) * sqrt(1.0);
}

Julia

function code(J, K, U)
	return Float64(fma(Float64(Float64(0.25 * J) * K), K, Float64(-2.0 * J)) * sqrt(1.0))
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(N[(0.25 * J), $MachinePrecision] * K), $MachinePrecision] * K + N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

2. Taylor expanded in J around inf

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in K around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 27.8%

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

7. Applied rewrites 27.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 27.8%

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

9. Applied rewrites 27.8%

   \[\leadsto \mathsf{fma}\left(\left(0.25 \cdot J\right) \cdot K, \color{blue}{K}, -2 \cdot J\right) \cdot \sqrt{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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)))))