Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.6% → 87.1%

Time: 7.7s

Alternatives: 13

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: 73.6% 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: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := -2 \cdot \left|J\right|\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ t_4 := t\_2 \cdot \left(-1 \cdot \left(\left|U\right| \cdot \left(-1 \cdot \frac{\sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{\left|J\right|}\right)\right)\right)\\ t_5 := \cos \left(K \cdot 0.5\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\left(t\_0 \cdot t\_5\right) \cdot \sqrt{1 + {\left(\frac{\left|U\right|}{\left(\left|J\right| + \left|J\right|\right) \cdot t\_5}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (fabs J)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* t_0 t_1))
        (t_3
         (*
          t_2
          (sqrt (+ 1.0 (pow (/ (fabs U) (* (* 2.0 (fabs J)) t_1)) 2.0)))))
        (t_4
         (*
          t_2
          (*
           -1.0
           (*
            (fabs U)
            (* -1.0 (/ (sqrt (/ 0.25 (pow (cos (* 0.5 K)) 2.0))) (fabs J)))))))
        (t_5 (cos (* K 0.5))))
   (*
    (copysign 1.0 J)
    (if (<= t_3 (- INFINITY))
      t_4
      (if (<= t_3 2e+287)
        (*
         (* t_0 t_5)
         (sqrt (+ 1.0 (pow (/ (fabs U) (* (+ (fabs J) (fabs J)) t_5)) 2.0))))
        t_4)))))

C

double code(double J, double K, double U) {
	double t_0 = -2.0 * fabs(J);
	double t_1 = cos((K / 2.0));
	double t_2 = t_0 * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((fabs(U) / ((2.0 * fabs(J)) * t_1)), 2.0)));
	double t_4 = t_2 * (-1.0 * (fabs(U) * (-1.0 * (sqrt((0.25 / pow(cos((0.5 * K)), 2.0))) / fabs(J)))));
	double t_5 = cos((K * 0.5));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= 2e+287) {
		tmp = (t_0 * t_5) * sqrt((1.0 + pow((fabs(U) / ((fabs(J) + fabs(J)) * t_5)), 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 = -2.0 * Math.abs(J);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = t_0 * t_1;
	double t_3 = t_2 * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((2.0 * Math.abs(J)) * t_1)), 2.0)));
	double t_4 = t_2 * (-1.0 * (Math.abs(U) * (-1.0 * (Math.sqrt((0.25 / Math.pow(Math.cos((0.5 * K)), 2.0))) / Math.abs(J)))));
	double t_5 = Math.cos((K * 0.5));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_4;
	} else if (t_3 <= 2e+287) {
		tmp = (t_0 * t_5) * Math.sqrt((1.0 + Math.pow((Math.abs(U) / ((Math.abs(J) + Math.abs(J)) * t_5)), 2.0)));
	} else {
		tmp = t_4;
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = -2.0 * math.fabs(J)
	t_1 = math.cos((K / 2.0))
	t_2 = t_0 * t_1
	t_3 = t_2 * math.sqrt((1.0 + math.pow((math.fabs(U) / ((2.0 * math.fabs(J)) * t_1)), 2.0)))
	t_4 = t_2 * (-1.0 * (math.fabs(U) * (-1.0 * (math.sqrt((0.25 / math.pow(math.cos((0.5 * K)), 2.0))) / math.fabs(J)))))
	t_5 = math.cos((K * 0.5))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_4
	elif t_3 <= 2e+287:
		tmp = (t_0 * t_5) * math.sqrt((1.0 + math.pow((math.fabs(U) / ((math.fabs(J) + math.fabs(J)) * t_5)), 2.0)))
	else:
		tmp = t_4
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(-2.0 * abs(J))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0))))
	t_4 = Float64(t_2 * Float64(-1.0 * Float64(abs(U) * Float64(-1.0 * Float64(sqrt(Float64(0.25 / (cos(Float64(0.5 * K)) ^ 2.0))) / abs(J))))))
	t_5 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= 2e+287)
		tmp = Float64(Float64(t_0 * t_5) * sqrt(Float64(1.0 + (Float64(abs(U) / Float64(Float64(abs(J) + abs(J)) * t_5)) ^ 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 = -2.0 * abs(J);
	t_1 = cos((K / 2.0));
	t_2 = t_0 * t_1;
	t_3 = t_2 * sqrt((1.0 + ((abs(U) / ((2.0 * abs(J)) * t_1)) ^ 2.0)));
	t_4 = t_2 * (-1.0 * (abs(U) * (-1.0 * (sqrt((0.25 / (cos((0.5 * K)) ^ 2.0))) / abs(J)))));
	t_5 = cos((K * 0.5));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_4;
	elseif (t_3 <= 2e+287)
		tmp = (t_0 * t_5) * sqrt((1.0 + ((abs(U) / ((abs(J) + abs(J)) * t_5)) ^ 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[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / 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[(t$95$2 * N[(-1.0 * N[(N[Abs[U], $MachinePrecision] * N[(-1.0 * N[(N[Sqrt[N[(0.25 / N[Power[N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(K * 0.5), $MachinePrecision]], $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, 2e+287], N[(N[(t$95$0 * t$95$5), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[Abs[U], $MachinePrecision] / N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$5), $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 2.0000000000000002e287 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.6%

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

   2. Taylor expanded in U around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 12.6

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(-1 \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}}\right)\right) \]

   4. Applied rewrites 12.6%

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

   5. Taylor expanded in J around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 20.7

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(-1 \cdot \left(U \cdot \left(-1 \cdot \frac{\sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}\right)\right)\right) \]

   7. Applied rewrites 20.7%

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

   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))))) < 2.0000000000000002e287

   1. Initial program 73.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.6

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

   3. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 73.6

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

   7. Applied rewrites 73.6%

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

Alternative 2: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \sqrt{\frac{0.25}{{J}^{2} \cdot {t\_0}^{2}}}\\ \mathbf{if}\;\left|U\right| \leq 3.2 \cdot 10^{+241}:\\ \;\;\;\;\left(t\_0 \cdot \cosh \sinh^{-1} \left(\frac{\left|U\right|}{\left(J + J\right) \cdot t\_0}\right)\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left|U\right| \cdot \mathsf{fma}\left(-2, J \cdot \left(t\_0 \cdot t\_1\right), -1 \cdot \frac{J \cdot t\_0}{{\left(\left|U\right|\right)}^{2} \cdot t\_1}\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (sqrt (/ 0.25 (* (pow J 2.0) (pow t_0 2.0))))))
   (if (<= (fabs U) 3.2e+241)
     (* (* t_0 (cosh (asinh (/ (fabs U) (* (+ J J) t_0))))) (* J -2.0))
     (*
      (fabs U)
      (fma
       -2.0
       (* J (* t_0 t_1))
       (* -1.0 (/ (* J t_0) (* (pow (fabs U) 2.0) t_1))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = sqrt((0.25 / (pow(J, 2.0) * pow(t_0, 2.0))));
	double tmp;
	if (fabs(U) <= 3.2e+241) {
		tmp = (t_0 * cosh(asinh((fabs(U) / ((J + J) * t_0))))) * (J * -2.0);
	} else {
		tmp = fabs(U) * fma(-2.0, (J * (t_0 * t_1)), (-1.0 * ((J * t_0) / (pow(fabs(U), 2.0) * t_1))));
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = sqrt(Float64(0.25 / Float64((J ^ 2.0) * (t_0 ^ 2.0))))
	tmp = 0.0
	if (abs(U) <= 3.2e+241)
		tmp = Float64(Float64(t_0 * cosh(asinh(Float64(abs(U) / Float64(Float64(J + J) * t_0))))) * Float64(J * -2.0));
	else
		tmp = Float64(abs(U) * fma(-2.0, Float64(J * Float64(t_0 * t_1)), Float64(-1.0 * Float64(Float64(J * t_0) / Float64((abs(U) ^ 2.0) * t_1)))));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.25 / N[(N[Power[J, 2.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[U], $MachinePrecision], 3.2e+241], N[(N[(t$95$0 * N[Cosh[N[ArcSinh[N[(N[Abs[U], $MachinePrecision] / N[(N[(J + J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Abs[U], $MachinePrecision] * N[(-2.0 * N[(J * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[(J * t$95$0), $MachinePrecision] / N[(N[Power[N[Abs[U], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if U < 3.20000000000000004e241

   1. Initial program 73.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.6

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

   3. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   7. Applied rewrites 84.7%

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

   if 3.20000000000000004e241 < U

   1. Initial program 73.6%

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

   2. Taylor expanded in U around inf

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

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

   4. Applied rewrites 14.2%

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

Alternative 3: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{0.25}{{J}^{2} \cdot \left(0.5 + 0.5 \cdot \cos K\right)}}\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := \cos \left(-0.5 \cdot K\right)\\ \mathbf{if}\;\left|U\right| \leq 3.2 \cdot 10^{+241}:\\ \;\;\;\;\left(t\_1 \cdot \cosh \sinh^{-1} \left(\frac{\left|U\right|}{\left(J + J\right) \cdot t\_1}\right)\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left|U\right| \cdot \mathsf{fma}\left(-2, J \cdot \left(t\_2 \cdot t\_0\right), -1 \cdot \frac{J \cdot t\_2}{{\left(\left|U\right|\right)}^{2} \cdot t\_0}\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.25 (* (pow J 2.0) (+ 0.5 (* 0.5 (cos K)))))))
        (t_1 (cos (* 0.5 K)))
        (t_2 (cos (* -0.5 K))))
   (if (<= (fabs U) 3.2e+241)
     (* (* t_1 (cosh (asinh (/ (fabs U) (* (+ J J) t_1))))) (* J -2.0))
     (*
      (fabs U)
      (fma
       -2.0
       (* J (* t_2 t_0))
       (* -1.0 (/ (* J t_2) (* (pow (fabs U) 2.0) t_0))))))))

C

double code(double J, double K, double U) {
	double t_0 = sqrt((0.25 / (pow(J, 2.0) * (0.5 + (0.5 * cos(K))))));
	double t_1 = cos((0.5 * K));
	double t_2 = cos((-0.5 * K));
	double tmp;
	if (fabs(U) <= 3.2e+241) {
		tmp = (t_1 * cosh(asinh((fabs(U) / ((J + J) * t_1))))) * (J * -2.0);
	} else {
		tmp = fabs(U) * fma(-2.0, (J * (t_2 * t_0)), (-1.0 * ((J * t_2) / (pow(fabs(U), 2.0) * t_0))));
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = sqrt(Float64(0.25 / Float64((J ^ 2.0) * Float64(0.5 + Float64(0.5 * cos(K))))))
	t_1 = cos(Float64(0.5 * K))
	t_2 = cos(Float64(-0.5 * K))
	tmp = 0.0
	if (abs(U) <= 3.2e+241)
		tmp = Float64(Float64(t_1 * cosh(asinh(Float64(abs(U) / Float64(Float64(J + J) * t_1))))) * Float64(J * -2.0));
	else
		tmp = Float64(abs(U) * fma(-2.0, Float64(J * Float64(t_2 * t_0)), Float64(-1.0 * Float64(Float64(J * t_2) / Float64((abs(U) ^ 2.0) * t_0)))));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Sqrt[N[(0.25 / N[(N[Power[J, 2.0], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[U], $MachinePrecision], 3.2e+241], N[(N[(t$95$1 * N[Cosh[N[ArcSinh[N[(N[Abs[U], $MachinePrecision] / N[(N[(J + J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Abs[U], $MachinePrecision] * N[(-2.0 * N[(J * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[(J * t$95$2), $MachinePrecision] / N[(N[Power[N[Abs[U], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if U < 3.20000000000000004e241

   1. Initial program 73.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.6

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

   3. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   7. Applied rewrites 84.7%

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

   if 3.20000000000000004e241 < U

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in U around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   6. Applied rewrites 14.2%

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

Alternative 4: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_0}\right)}^{2}}\\ t_3 := -2 \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U}{\left(\cos K - -1\right) \cdot \left(\left|J\right| + \left|J\right|\right)}, \frac{U}{\left|J\right|}, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* -2.0 (fabs J)))
        (t_2
         (*
          (* t_1 t_0)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_0)) 2.0)))))
        (t_3
         (*
          -2.0
          (*
           (cos (* -0.5 K))
           (sqrt (* 0.25 (/ (pow U 2.0) (+ 0.5 (* 0.5 (cos K))))))))))
   (*
    (copysign 1.0 J)
    (if (<= t_2 (- INFINITY))
      t_3
      (if (<= t_2 2e+287)
        (*
         (*
          (sqrt
           (fma
            (/ U (* (- (cos K) -1.0) (+ (fabs J) (fabs J))))
            (/ U (fabs J))
            1.0))
          (cos (* K -0.5)))
         t_1)
        t_3)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = -2.0 * fabs(J);
	double t_2 = (t_1 * t_0) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_0)), 2.0)));
	double t_3 = -2.0 * (cos((-0.5 * K)) * sqrt((0.25 * (pow(U, 2.0) / (0.5 + (0.5 * cos(K)))))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 2e+287) {
		tmp = (sqrt(fma((U / ((cos(K) - -1.0) * (fabs(J) + fabs(J)))), (U / fabs(J)), 1.0)) * cos((K * -0.5))) * t_1;
	} else {
		tmp = t_3;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(-2.0 * abs(J))
	t_2 = Float64(Float64(t_1 * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_0)) ^ 2.0))))
	t_3 = Float64(-2.0 * Float64(cos(Float64(-0.5 * K)) * sqrt(Float64(0.25 * Float64((U ^ 2.0) / Float64(0.5 + Float64(0.5 * cos(K))))))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 2e+287)
		tmp = Float64(Float64(sqrt(fma(Float64(U / Float64(Float64(cos(K) - -1.0) * Float64(abs(J) + abs(J)))), Float64(U / abs(J)), 1.0)) * cos(Float64(K * -0.5))) * t_1);
	else
		tmp = t_3;
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * 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]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 2e+287], N[(N[(N[Sqrt[N[(N[(U / N[(N[(N[Cos[K], $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$3]]), $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 2.0000000000000002e287 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in J around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 15.2

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

   6. Applied rewrites 15.2%

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

   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))))) < 2.0000000000000002e287

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 70.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 73.5%

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

Alternative 5: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 73.6

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

3. Applied rewrites 73.6%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 73.6

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

5. Applied rewrites 73.6%

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

7. Applied rewrites 84.7%

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

Alternative 6: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := 1 + -0.125 \cdot {K}^{2}\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := t\_3 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_5 := \left|J\right| + \left|J\right|\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_3 \cdot \cosh \sinh^{-1} \left(\frac{U}{t\_5}\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U}{\left(\cos K - -1\right) \cdot t\_5}, \frac{U}{\left|J\right|}, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot \left|J\right|\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{t\_5 \cdot t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.125 (pow K 2.0))))
        (t_1 (* -2.0 (fabs J)))
        (t_2 (cos (/ K 2.0)))
        (t_3 (* t_1 t_2))
        (t_4 (* t_3 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_2)) 2.0)))))
        (t_5 (+ (fabs J) (fabs J))))
   (*
    (copysign 1.0 J)
    (if (<= t_4 (- INFINITY))
      (* t_3 (cosh (asinh (/ U t_5))))
      (if (<= t_4 2e+287)
        (*
         (*
          (sqrt (fma (/ U (* (- (cos K) -1.0) t_5)) (/ U (fabs J)) 1.0))
          (cos (* K -0.5)))
         t_1)
        (* (* (* t_0 (fabs J)) -2.0) (cosh (asinh (/ U (* t_5 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 = -2.0 * fabs(J);
	double t_2 = cos((K / 2.0));
	double t_3 = t_1 * t_2;
	double t_4 = t_3 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_5 = fabs(J) + fabs(J);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_3 * cosh(asinh((U / t_5)));
	} else if (t_4 <= 2e+287) {
		tmp = (sqrt(fma((U / ((cos(K) - -1.0) * t_5)), (U / fabs(J)), 1.0)) * cos((K * -0.5))) * t_1;
	} else {
		tmp = ((t_0 * fabs(J)) * -2.0) * cosh(asinh((U / (t_5 * 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(-2.0 * abs(J))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(t_1 * t_2)
	t_4 = Float64(t_3 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_5 = Float64(abs(J) + abs(J))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(t_3 * cosh(asinh(Float64(U / t_5))));
	elseif (t_4 <= 2e+287)
		tmp = Float64(Float64(sqrt(fma(Float64(U / Float64(Float64(cos(K) - -1.0) * t_5)), Float64(U / abs(J)), 1.0)) * cos(Float64(K * -0.5))) * t_1);
	else
		tmp = Float64(Float64(Float64(t_0 * abs(J)) * -2.0) * cosh(asinh(Float64(U / Float64(t_5 * 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[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * 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]}, Block[{t$95$5 = N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $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[(U / t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+287], N[(N[(N[Sqrt[N[(N[(U / N[(N[(N[Cos[K], $MachinePrecision] - -1.0), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] * N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(t$95$0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U / N[(t$95$5 * 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 73.6%

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

      1. lift-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 84.7

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

      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 84.7

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

      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 84.7

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

   3. Applied rewrites 84.7%

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

   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 71.5

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

   6. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip-rev N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 71.5

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

   8. Applied rewrites 71.5%

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

   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))))) < 2.0000000000000002e287

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 70.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 73.5%

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

   if 2.0000000000000002e287 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 84.7

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

      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 84.7

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

      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 84.7

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

   3. Applied rewrites 84.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)} \]
   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 84.7

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

      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. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      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. lift-*.f64 N/A

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

      17. distribute-lft-neg-out N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{-1}{2} \cdot K\right)}\right) \]
   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 43.8

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

   8. Applied rewrites 43.8%

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

   9. Taylor expanded in K around 0

      \[\leadsto \left(\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}}\right) \]
   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 46.6

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

   11. Applied rewrites 46.6%

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

Alternative 7: 71.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * cos((K / 2.0))) * cosh(asinh((U / (J + 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[(U / N[(J + J), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.6%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. 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 84.7

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

   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 84.7

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

   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 84.7

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

3. Applied rewrites 84.7%

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

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 71.5

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

6. Applied rewrites 71.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. mult-flip-rev N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. *-commutative N/A

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

   8. count-2-rev N/A

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

   9. lift-+.f64 N/A

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

   10. lower-/.f64 71.5

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

8. Applied rewrites 71.5%

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

Alternative 8: 71.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. 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 84.7

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

   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 84.7

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

   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 84.7

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

3. Applied rewrites 84.7%

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

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 71.5

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

6. Applied rewrites 71.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 71.5

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

8. Applied rewrites 71.5%

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

Alternative 9: 64.7% accurate, 1.8× speedup

Math

\[\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U}{J} \cdot \left(\frac{U}{J} \cdot 0.25\right) - -1} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* (cos (* -0.5 K)) -2.0) J)
  (sqrt (- (* (/ U J) (* (/ U J) 0.25)) -1.0))))

C

double code(double J, double K, double U) {
	return ((cos((-0.5 * K)) * -2.0) * J) * sqrt((((U / J) * ((U / J) * 0.25)) - -1.0));
}

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 = ((cos(((-0.5d0) * k)) * (-2.0d0)) * j) * sqrt((((u / j) * ((u / j) * 0.25d0)) - (-1.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	return ((Math.cos((-0.5 * K)) * -2.0) * J) * Math.sqrt((((U / J) * ((U / J) * 0.25)) - -1.0));
}

Python

def code(J, K, U):
	return ((math.cos((-0.5 * K)) * -2.0) * J) * math.sqrt((((U / J) * ((U / J) * 0.25)) - -1.0))

Julia

function code(J, K, U)
	return Float64(Float64(Float64(cos(Float64(-0.5 * K)) * -2.0) * J) * sqrt(Float64(Float64(Float64(U / J) * Float64(Float64(U / J) * 0.25)) - -1.0)))
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((cos((-0.5 * K)) * -2.0) * J) * sqrt((((U / J) * ((U / J) * 0.25)) - -1.0));
end

Wolfram

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

Derivation

1. Initial program 73.6%

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

3. Applied rewrites 73.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/l/ N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 70.4

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 70.4

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

5. Applied rewrites 70.4%

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

6. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 51.8

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

8. Applied rewrites 51.8%

   \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{0.25 \cdot \frac{{U}^{2}}{{J}^{2}}} - -1} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. frac-times N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. mult-flip N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/l* N/A

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

   16. lower-*.f64 N/A

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

   17. mult-flip N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 64.7

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

10. Applied rewrites 64.7%

    \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U}{J} \cdot \color{blue}{\left(\frac{U}{J} \cdot 0.25\right)} - -1} \]
11. Add Preprocessing

Alternative 10: 56.8% accurate, 1.8× speedup

Math

\[\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{\left(U \cdot \frac{U}{J \cdot J}\right) \cdot 0.25 - -1} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* (cos (* 0.5 K)) J) -2.0)
  (sqrt (- (* (* U (/ U (* J J))) 0.25) -1.0))))

C

double code(double J, double K, double U) {
	return ((cos((0.5 * K)) * J) * -2.0) * sqrt((((U * (U / (J * J))) * 0.25) - -1.0));
}

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 = ((cos((0.5d0 * k)) * j) * (-2.0d0)) * sqrt((((u * (u / (j * j))) * 0.25d0) - (-1.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	return ((Math.cos((0.5 * K)) * J) * -2.0) * Math.sqrt((((U * (U / (J * J))) * 0.25) - -1.0));
}

Python

def code(J, K, U):
	return ((math.cos((0.5 * K)) * J) * -2.0) * math.sqrt((((U * (U / (J * J))) * 0.25) - -1.0))

Julia

function code(J, K, U)
	return Float64(Float64(Float64(cos(Float64(0.5 * K)) * J) * -2.0) * sqrt(Float64(Float64(Float64(U * Float64(U / Float64(J * J))) * 0.25) - -1.0)))
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((cos((0.5 * K)) * J) * -2.0) * sqrt((((U * (U / (J * J))) * 0.25) - -1.0));
end

Wolfram

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

Derivation

1. Initial program 73.6%

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

3. Applied rewrites 73.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/l/ N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 70.4

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 70.4

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

5. Applied rewrites 70.4%

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

6. Taylor expanded in K around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 51.8

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

8. Applied rewrites 51.8%

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

10. Applied rewrites 56.8%

    \[\leadsto \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{\left(U \cdot \frac{U}{J \cdot J}\right) \cdot 0.25 - -1}} \]
11. Add Preprocessing

Alternative 11: 53.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|K\right| \leq 40000:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{J}^{2}} - -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot \left|K\right|\right) \cdot J\right) \cdot -2\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= (fabs K) 40000.0)
   (* (* -2.0 J) (sqrt (- (* 0.25 (/ (pow U 2.0) (pow J 2.0))) -1.0)))
   (* (* (* (cos (* -0.5 (fabs K))) J) -2.0) 1.0)))

C

double code(double J, double K, double U) {
	double tmp;
	if (fabs(K) <= 40000.0) {
		tmp = (-2.0 * J) * sqrt(((0.25 * (pow(U, 2.0) / pow(J, 2.0))) - -1.0));
	} else {
		tmp = ((cos((-0.5 * fabs(K))) * J) * -2.0) * 1.0;
	}
	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) <= 40000.0d0) then
        tmp = ((-2.0d0) * j) * sqrt(((0.25d0 * ((u ** 2.0d0) / (j ** 2.0d0))) - (-1.0d0)))
    else
        tmp = ((cos(((-0.5d0) * abs(k))) * j) * (-2.0d0)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (Math.abs(K) <= 40000.0) {
		tmp = (-2.0 * J) * Math.sqrt(((0.25 * (Math.pow(U, 2.0) / Math.pow(J, 2.0))) - -1.0));
	} else {
		tmp = ((Math.cos((-0.5 * Math.abs(K))) * J) * -2.0) * 1.0;
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if math.fabs(K) <= 40000.0:
		tmp = (-2.0 * J) * math.sqrt(((0.25 * (math.pow(U, 2.0) / math.pow(J, 2.0))) - -1.0))
	else:
		tmp = ((math.cos((-0.5 * math.fabs(K))) * J) * -2.0) * 1.0
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (abs(K) <= 40000.0)
		tmp = Float64(Float64(-2.0 * J) * sqrt(Float64(Float64(0.25 * Float64((U ^ 2.0) / (J ^ 2.0))) - -1.0)));
	else
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * abs(K))) * J) * -2.0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (abs(K) <= 40000.0)
		tmp = (-2.0 * J) * sqrt(((0.25 * ((U ^ 2.0) / (J ^ 2.0))) - -1.0));
	else
		tmp = ((cos((-0.5 * abs(K))) * J) * -2.0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[N[Abs[K], $MachinePrecision], 40000.0], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / N[Power[J, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[N[(-0.5 * N[Abs[K], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 4e4

   1. Initial program 73.6%

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

   3. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 70.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 51.8

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

   8. Applied rewrites 51.8%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 32.9%

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

   if 4e4 < K

   1. Initial program 73.6%

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

   2. Taylor expanded in J around inf

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

   4. Applied rewrites 52.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 52.5

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

      8. lift-cos.f64 N/A

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

      9. cos-neg-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-lft-neg-out N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 52.5

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

   6. Applied rewrites 52.5%

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

Alternative 12: 52.5% accurate, 2.5× speedup

Math

\[\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot 1 \]

FPCore

(FPCore (J K U) :precision binary64 (* (* (* (cos (* -0.5 K)) J) -2.0) 1.0))

C

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

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 = ((cos(((-0.5d0) * k)) * j) * (-2.0d0)) * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

2. Taylor expanded in J around inf

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

4. Applied rewrites 52.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 52.5

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

   8. lift-cos.f64 N/A

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

   9. cos-neg-rev N/A

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

   10. lower-cos.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. mult-flip N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-lft-neg-out N/A

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

   18. metadata-eval N/A

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

   19. lift-*.f64 52.5

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

6. Applied rewrites 52.5%

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

Alternative 13: 28.0% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(J, K, U)
	return Float64(fma(Float64(Float64(0.25 * J) * K), K, Float64(-2.0 * J)) * 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] * 1.0), $MachinePrecision]

Derivation

1. Initial program 73.6%

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

2. Taylor expanded in J around inf

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

4. Applied rewrites 52.5%

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

5. Taylor expanded in K around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 28.0

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

7. Applied rewrites 28.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 28.0

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

9. Applied rewrites 28.0%

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

Reproduce

herbie shell --seed 2025175 
(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)))))