Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 88.3%

Time: 9.4s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))))
   (* (* (hypot (/ U (* (+ J J) t_0)) 1.0) -2.0) (* t_0 J))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K * 0.5));
	return (hypot((U / ((J + J) * t_0)), 1.0) * -2.0) * (t_0 * J);
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	return (Math.hypot((U / ((J + J) * t_0)), 1.0) * -2.0) * (t_0 * J);
}

Python

def code(J, K, U):
	t_0 = math.cos((K * 0.5))
	return (math.hypot((U / ((J + J) * t_0)), 1.0) * -2.0) * (t_0 * J)

Julia

function code(J, K, U)
	t_0 = cos(Float64(K * 0.5))
	return Float64(Float64(hypot(Float64(U / Float64(Float64(J + J) * t_0)), 1.0) * -2.0) * Float64(t_0 * J))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K * 0.5));
	tmp = (hypot((U / ((J + J) * t_0)), 1.0) * -2.0) * (t_0 * J);
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Sqrt[N[(U / N[(N[(J + J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2 + 1.0 ^ 2], $MachinePrecision] * -2.0), $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 73.3%

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

3. Applied rewrites 73.3%

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

5. Applied rewrites 88.3%

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

7. Applied rewrites 88.3%

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

Alternative 2: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ t_1 := t\_0 \cdot \frac{U \cdot \sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos K}}}{J}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(-2 \cdot J\right) \cdot t\_2\\ t_4 := t\_3 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_2}\right)}^{2}}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;t\_3 \cdot \sqrt{1 + \frac{U \cdot 0.25}{J} \cdot \frac{U}{J}}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot \left(J \cdot \left(\left(\cos K - -1\right) \cdot 2\right)\right)}, U, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (* -2.0 J) (cos (* K 0.5))))
        (t_1 (* t_0 (/ (* U (sqrt (/ 0.25 (+ 0.5 (* 0.5 (cos K)))))) J)))
        (t_2 (cos (/ K 2.0)))
        (t_3 (* (* -2.0 J) t_2))
        (t_4 (* t_3 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_2)) 2.0))))))
   (if (<= t_4 (- INFINITY))
     t_1
     (if (<= t_4 2e-87)
       (* t_3 (sqrt (+ 1.0 (* (/ (* U 0.25) J) (/ U J)))))
       (if (<= t_4 5e+307)
         (* t_0 (sqrt (fma (/ U (* J (* J (* (- (cos K) -1.0) 2.0)))) U 1.0)))
         t_1)))))

C

double code(double J, double K, double U) {
	double t_0 = (-2.0 * J) * cos((K * 0.5));
	double t_1 = t_0 * ((U * sqrt((0.25 / (0.5 + (0.5 * cos(K)))))) / J);
	double t_2 = cos((K / 2.0));
	double t_3 = (-2.0 * J) * t_2;
	double t_4 = t_3 * sqrt((1.0 + pow((U / ((2.0 * J) * t_2)), 2.0)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_4 <= 2e-87) {
		tmp = t_3 * sqrt((1.0 + (((U * 0.25) / J) * (U / J))));
	} else if (t_4 <= 5e+307) {
		tmp = t_0 * sqrt(fma((U / (J * (J * ((cos(K) - -1.0) * 2.0)))), U, 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	t_1 = Float64(t_0 * Float64(Float64(U * sqrt(Float64(0.25 / Float64(0.5 + Float64(0.5 * cos(K)))))) / J))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(-2.0 * J) * t_2)
	t_4 = Float64(t_3 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_4 <= 2e-87)
		tmp = Float64(t_3 * sqrt(Float64(1.0 + Float64(Float64(Float64(U * 0.25) / J) * Float64(U / J)))));
	elseif (t_4 <= 5e+307)
		tmp = Float64(t_0 * sqrt(fma(Float64(U / Float64(J * Float64(J * Float64(Float64(cos(K) - -1.0) * 2.0)))), U, 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(U * N[Sqrt[N[(0.25 / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$1, If[LessEqual[t$95$4, 2e-87], N[(t$95$3 * N[Sqrt[N[(1.0 + N[(N[(N[(U * 0.25), $MachinePrecision] / J), $MachinePrecision] * N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e+307], N[(t$95$0 * N[Sqrt[N[(N[(U / N[(J * N[(J * N[(N[(N[Cos[K], $MachinePrecision] - -1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 5e307 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 13.3%

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

   6. Applied rewrites 13.3%

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

   7. Taylor expanded in J around -inf

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

   9. Applied rewrites 20.7%

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

   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))))) < 2.00000000000000004e-87

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 64.8%

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

   if 2.00000000000000004e-87 < (*.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))))) < 5e307

   1. Initial program 73.3%

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

   3. Applied rewrites 73.3%

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

   5. Applied rewrites 61.7%

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

   7. Applied rewrites 61.7%

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

Alternative 3: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_3 := \cos \left(K \cdot 0.5\right)\\ t_4 := \left(\left(-2 \cdot J\right) \cdot t\_3\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos K}}}{J}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + \frac{U \cdot 0.25}{J} \cdot \frac{U}{J}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U}{\left(\left(\cos K - -1\right) \cdot 2\right) \cdot \left(J \cdot J\right)}, U, 1\right)} \cdot t\_3\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0)))))
        (t_3 (cos (* K 0.5)))
        (t_4
         (*
          (* (* -2.0 J) t_3)
          (/ (* U (sqrt (/ 0.25 (+ 0.5 (* 0.5 (cos K)))))) J))))
   (if (<= t_2 (- INFINITY))
     t_4
     (if (<= t_2 2e-87)
       (* t_1 (sqrt (+ 1.0 (* (/ (* U 0.25) J) (/ U J)))))
       (if (<= t_2 5e+307)
         (*
          (* (sqrt (fma (/ U (* (* (- (cos K) -1.0) 2.0) (* J J))) U 1.0)) t_3)
          (* -2.0 J))
         t_4)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
	double t_3 = cos((K * 0.5));
	double t_4 = ((-2.0 * J) * t_3) * ((U * sqrt((0.25 / (0.5 + (0.5 * cos(K)))))) / J);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_2 <= 2e-87) {
		tmp = t_1 * sqrt((1.0 + (((U * 0.25) / J) * (U / J))));
	} else if (t_2 <= 5e+307) {
		tmp = (sqrt(fma((U / (((cos(K) - -1.0) * 2.0) * (J * J))), U, 1.0)) * t_3) * (-2.0 * J);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_3 = cos(Float64(K * 0.5))
	t_4 = Float64(Float64(Float64(-2.0 * J) * t_3) * Float64(Float64(U * sqrt(Float64(0.25 / Float64(0.5 + Float64(0.5 * cos(K)))))) / J))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_2 <= 2e-87)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + Float64(Float64(Float64(U * 0.25) / J) * Float64(U / J)))));
	elseif (t_2 <= 5e+307)
		tmp = Float64(Float64(sqrt(fma(Float64(U / Float64(Float64(Float64(cos(K) - -1.0) * 2.0) * Float64(J * J))), U, 1.0)) * t_3) * Float64(-2.0 * J));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 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]}, Block[{t$95$3 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(N[(U * N[Sqrt[N[(0.25 / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$4, If[LessEqual[t$95$2, 2e-87], N[(t$95$1 * N[Sqrt[N[(1.0 + N[(N[(N[(U * 0.25), $MachinePrecision] / J), $MachinePrecision] * N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+307], N[(N[(N[Sqrt[N[(N[(U / N[(N[(N[(N[Cos[K], $MachinePrecision] - -1.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 5e307 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 13.3%

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

   6. Applied rewrites 13.3%

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

   7. Taylor expanded in J around -inf

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

   9. Applied rewrites 20.7%

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

   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))))) < 2.00000000000000004e-87

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 64.8%

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

   if 2.00000000000000004e-87 < (*.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))))) < 5e307

   1. Initial program 73.3%

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

   3. Applied rewrites 73.3%

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

   5. Applied rewrites 61.7%

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

   7. Applied rewrites 61.7%

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

Alternative 4: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \frac{U \cdot \sqrt{\frac{0.25}{0.5 + 0.5 \cdot \cos K}}}{J}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + \frac{U \cdot 0.25}{J} \cdot \frac{U}{J}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0
         (*
          (* (* -2.0 J) (cos (* K 0.5)))
          (/ (* U (sqrt (/ 0.25 (+ 0.5 (* 0.5 (cos K)))))) J)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_1)) 2.0))))))
   (if (<= t_3 (- INFINITY))
     t_0
     (if (<= t_3 5e+307)
       (* t_2 (sqrt (+ 1.0 (* (/ (* U 0.25) J) (/ U J)))))
       t_0))))

C

double code(double J, double K, double U) {
	double t_0 = ((-2.0 * J) * cos((K * 0.5))) * ((U * sqrt((0.25 / (0.5 + (0.5 * cos(K)))))) / J);
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_3 <= 5e+307) {
		tmp = t_2 * sqrt((1.0 + (((U * 0.25) / J) * (U / J))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = ((-2.0 * J) * Math.cos((K * 0.5))) * ((U * Math.sqrt((0.25 / (0.5 + (0.5 * Math.cos(K)))))) / J);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = (-2.0 * J) * t_1;
	double t_3 = t_2 * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_3 <= 5e+307) {
		tmp = t_2 * Math.sqrt((1.0 + (((U * 0.25) / J) * (U / J))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = ((-2.0 * J) * math.cos((K * 0.5))) * ((U * math.sqrt((0.25 / (0.5 + (0.5 * math.cos(K)))))) / J)
	t_1 = math.cos((K / 2.0))
	t_2 = (-2.0 * J) * t_1
	t_3 = t_2 * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_1)), 2.0)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_0
	elif t_3 <= 5e+307:
		tmp = t_2 * math.sqrt((1.0 + (((U * 0.25) / J) * (U / J))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5))) * Float64(Float64(U * sqrt(Float64(0.25 / Float64(0.5 + Float64(0.5 * cos(K)))))) / J))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_3 <= 5e+307)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + Float64(Float64(Float64(U * 0.25) / J) * Float64(U / J)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = ((-2.0 * J) * cos((K * 0.5))) * ((U * sqrt((0.25 / (0.5 + (0.5 * cos(K)))))) / J);
	t_1 = cos((K / 2.0));
	t_2 = (-2.0 * J) * t_1;
	t_3 = t_2 * sqrt((1.0 + ((U / ((2.0 * J) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_0;
	elseif (t_3 <= 5e+307)
		tmp = t_2 * sqrt((1.0 + (((U * 0.25) / J) * (U / J))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(U * N[Sqrt[N[(0.25 / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$0, If[LessEqual[t$95$3, 5e+307], N[(t$95$2 * N[Sqrt[N[(1.0 + N[(N[(N[(U * 0.25), $MachinePrecision] / J), $MachinePrecision] * N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

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 5e307 < (*.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 73.3%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 13.3%

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

   6. Applied rewrites 13.3%

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

   7. Taylor expanded in J around -inf

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

   9. Applied rewrites 20.7%

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

   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))))) < 5e307

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 64.8%

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

Alternative 5: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -0.125 \cdot {K}^{2}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(0.5 \cdot \frac{U}{J}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{hypot}\left(\frac{U}{\left(J + J\right) \cdot t\_0}, 1\right) \cdot -2\right) \cdot \left(t\_0 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.125 (pow K 2.0)))) (t_1 (cos (/ K 2.0))))
   (if (<=
        (* (* (* -2.0 J) t_1) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_1)) 2.0))))
        5e+307)
     (* (* (* -2.0 J) (cos (* K 0.5))) (hypot (* 0.5 (/ U J)) 1.0))
     (* (* (hypot (/ U (* (+ J J) t_0)) 1.0) -2.0) (* t_0 J)))))

C

double code(double J, double K, double U) {
	double t_0 = 1.0 + (-0.125 * pow(K, 2.0));
	double t_1 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_1) * sqrt((1.0 + pow((U / ((2.0 * J) * t_1)), 2.0)))) <= 5e+307) {
		tmp = ((-2.0 * J) * cos((K * 0.5))) * hypot((0.5 * (U / J)), 1.0);
	} else {
		tmp = (hypot((U / ((J + J) * t_0)), 1.0) * -2.0) * (t_0 * J);
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = 1.0 + (-0.125 * Math.pow(K, 2.0));
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_1) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_1)), 2.0)))) <= 5e+307) {
		tmp = ((-2.0 * J) * Math.cos((K * 0.5))) * Math.hypot((0.5 * (U / J)), 1.0);
	} else {
		tmp = (Math.hypot((U / ((J + J) * t_0)), 1.0) * -2.0) * (t_0 * J);
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = 1.0 + (-0.125 * math.pow(K, 2.0))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J) * t_1) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_1)), 2.0)))) <= 5e+307:
		tmp = ((-2.0 * J) * math.cos((K * 0.5))) * math.hypot((0.5 * (U / J)), 1.0)
	else:
		tmp = (math.hypot((U / ((J + J) * t_0)), 1.0) * -2.0) * (t_0 * J)
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(1.0 + Float64(-0.125 * (K ^ 2.0)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_1)) ^ 2.0)))) <= 5e+307)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5))) * hypot(Float64(0.5 * Float64(U / J)), 1.0));
	else
		tmp = Float64(Float64(hypot(Float64(U / Float64(Float64(J + J) * t_0)), 1.0) * -2.0) * Float64(t_0 * J));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = 1.0 + (-0.125 * (K ^ 2.0));
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J) * t_1) * sqrt((1.0 + ((U / ((2.0 * J) * t_1)) ^ 2.0)))) <= 5e+307)
		tmp = ((-2.0 * J) * cos((K * 0.5))) * hypot((0.5 * (U / J)), 1.0);
	else
		tmp = (hypot((U / ((J + J) * t_0)), 1.0) * -2.0) * (t_0 * J);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(1.0 + N[(-0.125 * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+307], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2 + 1.0 ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(U / N[(N[(J + J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2 + 1.0 ^ 2], $MachinePrecision] * -2.0), $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $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))))) < 5e307

   1. Initial program 73.3%

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

   3. Applied rewrites 73.3%

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 73.7%

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

   if 5e307 < (*.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 73.3%

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

   3. Applied rewrites 73.3%

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

   5. Applied rewrites 88.3%

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in K around 0

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

   10. Applied rewrites 70.9%

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

   11. Taylor expanded in K around 0

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

   13. Applied rewrites 49.9%

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

Alternative 6: 71.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(0.5 \cdot \frac{U}{J}, 1\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2 + 1.0 ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

3. Applied rewrites 73.3%

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

5. Applied rewrites 88.3%

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

6. Taylor expanded in K around 0

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

8. Applied rewrites 73.7%

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

Alternative 7: 65.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + \frac{U \cdot 0.25}{J} \cdot \frac{U}{J}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J + J\right) \cdot t\_0}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (fma (* K K) -0.125 1.0))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J) t_1)))
   (if (<= (* t_2 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_1)) 2.0)))) 5e+307)
     (* t_2 (sqrt (+ 1.0 (* (/ (* U 0.25) J) (/ U J)))))
     (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (+ J J) t_0)) 2.0)))))))

C

double code(double J, double K, double U) {
	double t_0 = fma((K * K), -0.125, 1.0);
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J) * t_1;
	double tmp;
	if ((t_2 * sqrt((1.0 + pow((U / ((2.0 * J) * t_1)), 2.0)))) <= 5e+307) {
		tmp = t_2 * sqrt((1.0 + (((U * 0.25) / J) * (U / J))));
	} else {
		tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((J + J) * t_0)), 2.0)));
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = fma(Float64(K * K), -0.125, 1.0)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J) * t_1)
	tmp = 0.0
	if (Float64(t_2 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_1)) ^ 2.0)))) <= 5e+307)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + Float64(Float64(Float64(U * 0.25) / J) * Float64(U / J)))));
	else
		tmp = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(J + J) * t_0)) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+307], N[(t$95$2 * N[Sqrt[N[(1.0 + N[(N[(N[(U * 0.25), $MachinePrecision] / J), $MachinePrecision] * N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(J + J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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))))) < 5e307

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 64.8%

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

   if 5e307 < (*.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 73.3%

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

   3. Applied rewrites 73.3%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 39.2%

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

   7. Taylor expanded in K around 0

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

   9. Applied rewrites 41.2%

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

   11. Applied rewrites 41.2%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{-0.125}, 1\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J + J\right) \cdot \left(1 + -0.125 \cdot {K}^{2}\right)}\right)}^{2}} \]

   13. Applied rewrites 41.2%

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

Alternative 8: 52.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\\ \mathbf{if}\;K \leq 0.0128:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J + J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (fma (* K K) -0.125 1.0)))
   (if (<= K 0.0128)
     (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (+ J J) t_0)) 2.0))))
     (* -2.0 (* (cos (* K 0.5)) J)))))

C

double code(double J, double K, double U) {
	double t_0 = fma((K * K), -0.125, 1.0);
	double tmp;
	if (K <= 0.0128) {
		tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((J + J) * t_0)), 2.0)));
	} else {
		tmp = -2.0 * (cos((K * 0.5)) * J);
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = fma(Float64(K * K), -0.125, 1.0)
	tmp = 0.0
	if (K <= 0.0128)
		tmp = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(J + J) * t_0)) ^ 2.0))));
	else
		tmp = Float64(-2.0 * Float64(cos(Float64(K * 0.5)) * J));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]}, If[LessEqual[K, 0.0128], N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(J + J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if K < 0.0128000000000000006

   1. Initial program 73.3%

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

   3. Applied rewrites 73.3%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 39.2%

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

   7. Taylor expanded in K around 0

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

   9. Applied rewrites 41.2%

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

   11. Applied rewrites 41.2%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{-0.125}, 1\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J + J\right) \cdot \left(1 + -0.125 \cdot {K}^{2}\right)}\right)}^{2}} \]

   13. Applied rewrites 41.2%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J + J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{-0.125}, 1\right)}\right)}^{2}} \]

   if 0.0128000000000000006 < K

   1. Initial program 73.3%

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

   3. Applied rewrites 73.3%

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

   5. Applied rewrites 88.3%

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in J around inf

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

   10. Applied rewrites 52.4%

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

Alternative 9: 52.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 6.6 \cdot 10^{-20}:\\ \;\;\;\;-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= K 6.6e-20)
   (* -2.0 (* J (sqrt (+ 1.0 (* 0.25 (/ (pow U 2.0) (pow J 2.0)))))))
   (* -2.0 (* (cos (* K 0.5)) J))))

C

double code(double J, double K, double U) {
	double tmp;
	if (K <= 6.6e-20) {
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * (pow(U, 2.0) / pow(J, 2.0))))));
	} else {
		tmp = -2.0 * (cos((K * 0.5)) * J);
	}
	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 (k <= 6.6d-20) then
        tmp = (-2.0d0) * (j * sqrt((1.0d0 + (0.25d0 * ((u ** 2.0d0) / (j ** 2.0d0))))))
    else
        tmp = (-2.0d0) * (cos((k * 0.5d0)) * j)
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (K <= 6.6e-20) {
		tmp = -2.0 * (J * Math.sqrt((1.0 + (0.25 * (Math.pow(U, 2.0) / Math.pow(J, 2.0))))));
	} else {
		tmp = -2.0 * (Math.cos((K * 0.5)) * J);
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if K <= 6.6e-20:
		tmp = -2.0 * (J * math.sqrt((1.0 + (0.25 * (math.pow(U, 2.0) / math.pow(J, 2.0))))))
	else:
		tmp = -2.0 * (math.cos((K * 0.5)) * J)
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (K <= 6.6e-20)
		tmp = Float64(-2.0 * Float64(J * sqrt(Float64(1.0 + Float64(0.25 * Float64((U ^ 2.0) / (J ^ 2.0)))))));
	else
		tmp = Float64(-2.0 * Float64(cos(Float64(K * 0.5)) * J));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (K <= 6.6e-20)
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * ((U ^ 2.0) / (J ^ 2.0))))));
	else
		tmp = -2.0 * (cos((K * 0.5)) * J);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[K, 6.6e-20], N[(-2.0 * N[(J * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / N[Power[J, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 6.6e-20

   1. Initial program 73.3%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 33.6%

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

   if 6.6e-20 < K

   1. Initial program 73.3%

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

   3. Applied rewrites 73.3%

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

   5. Applied rewrites 88.3%

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in J around inf

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

   10. Applied rewrites 52.4%

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

Alternative 10: 43.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot J\right) \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = (-2.0d0) * (cos((k * 0.5d0)) * j)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Applied rewrites 73.3%

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

5. Applied rewrites 88.3%

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

7. Applied rewrites 88.3%

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

8. Taylor expanded in J around inf

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

10. Applied rewrites 52.4%

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

Alternative 11: 28.6% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Applied rewrites 73.3%

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

4. Taylor expanded in J around inf

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

6. Applied rewrites 52.3%

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

7. Taylor expanded in K around 0

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

9. Applied rewrites 28.6%

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

11. Applied rewrites 28.6%

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

Alternative 12: 28.5% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Applied rewrites 73.3%

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

4. Taylor expanded in J around inf

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

6. Applied rewrites 52.3%

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

7. Taylor expanded in K around 0

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

9. Applied rewrites 28.6%

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

11. Applied rewrites 28.5%

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

Reproduce

herbie shell --seed 2025161 
(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)))))