Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.8% → 84.9%

Time: 5.0s

Alternatives: 11

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% 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: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}}\\ t_3 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(t\_3 \cdot \left(-1 \cdot \left(U \cdot \sqrt{\frac{0.25}{{t\_3}^{2}}}\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J + J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J) t_1)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_1)) 2.0)))))
        (t_3 (cos (* 0.5 K))))
   (if (<= t_2 (- INFINITY))
     (* -2.0 (* t_3 (* -1.0 (* U (sqrt (/ 0.25 (pow t_3 2.0)))))))
     (if (<= t_2 5e+302)
       (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (+ J J) t_0)) 2.0))))
       (* -2.0 (* -0.5 U))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + pow((U / ((2.0 * J) * t_1)), 2.0)));
	double t_3 = cos((0.5 * K));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (t_3 * (-1.0 * (U * sqrt((0.25 / pow(t_3, 2.0))))));
	} else if (t_2 <= 5e+302) {
		tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((J + J) * t_0)), 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_1)), 2.0)));
	double t_3 = Math.cos((0.5 * K));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (t_3 * (-1.0 * (U * Math.sqrt((0.25 / Math.pow(t_3, 2.0))))));
	} else if (t_2 <= 5e+302) {
		tmp = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((J + J) * t_0)), 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J) * t_1) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_1)), 2.0)))
	t_3 = math.cos((0.5 * K))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -2.0 * (t_3 * (-1.0 * (U * math.sqrt((0.25 / math.pow(t_3, 2.0))))))
	elif t_2 <= 5e+302:
		tmp = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((J + J) * t_0)), 2.0)))
	else:
		tmp = -2.0 * (-0.5 * U)
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_1)) ^ 2.0))))
	t_3 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(t_3 * Float64(-1.0 * Float64(U * sqrt(Float64(0.25 / (t_3 ^ 2.0)))))));
	elseif (t_2 <= 5e+302)
		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(-0.5 * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K * 0.5));
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + ((U / ((2.0 * J) * t_1)) ^ 2.0)));
	t_3 = cos((0.5 * K));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -2.0 * (t_3 * (-1.0 * (U * sqrt((0.25 / (t_3 ^ 2.0))))));
	elseif (t_2 <= 5e+302)
		tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((J + J) * t_0)) ^ 2.0)));
	else
		tmp = -2.0 * (-0.5 * U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J), $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]}, Block[{t$95$3 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(t$95$3 * N[(-1.0 * N[(U * N[Sqrt[N[(0.25 / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+302], 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[(-0.5 * U), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in U around -inf

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

   7. Applied rewrites 26.4%

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

   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))))) < 5e302

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   if 5e302 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in U around -inf

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

   10. Applied rewrites 27.0%

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

Alternative 2: 76.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(t\_2 \cdot \left(-1 \cdot \left(U \cdot \sqrt{\frac{0.25}{{t\_2}^{2}}}\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0)))))
        (t_2 (cos (* 0.5 K))))
   (if (<= t_1 (- INFINITY))
     (* -2.0 (* t_2 (* -1.0 (* U (sqrt (/ 0.25 (pow t_2 2.0)))))))
     (if (<= t_1 5e+302)
       (*
        (* (* -2.0 J) (cos (* K 0.5)))
        (sqrt (+ 1.0 (pow (* 0.5 (/ U J)) 2.0))))
       (* -2.0 (* -0.5 U))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
	double t_2 = cos((0.5 * K));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -2.0 * (t_2 * (-1.0 * (U * sqrt((0.25 / pow(t_2, 2.0))))));
	} else if (t_1 <= 5e+302) {
		tmp = ((-2.0 * J) * cos((K * 0.5))) * sqrt((1.0 + pow((0.5 * (U / J)), 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
	double t_2 = Math.cos((0.5 * K));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (t_2 * (-1.0 * (U * Math.sqrt((0.25 / Math.pow(t_2, 2.0))))));
	} else if (t_1 <= 5e+302) {
		tmp = ((-2.0 * J) * Math.cos((K * 0.5))) * Math.sqrt((1.0 + Math.pow((0.5 * (U / J)), 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))
	t_2 = math.cos((0.5 * K))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -2.0 * (t_2 * (-1.0 * (U * math.sqrt((0.25 / math.pow(t_2, 2.0))))))
	elif t_1 <= 5e+302:
		tmp = ((-2.0 * J) * math.cos((K * 0.5))) * math.sqrt((1.0 + math.pow((0.5 * (U / J)), 2.0)))
	else:
		tmp = -2.0 * (-0.5 * U)
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(t_2 * Float64(-1.0 * Float64(U * sqrt(Float64(0.25 / (t_2 ^ 2.0)))))));
	elseif (t_1 <= 5e+302)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U / J)) ^ 2.0))));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
	t_2 = cos((0.5 * K));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -2.0 * (t_2 * (-1.0 * (U * sqrt((0.25 / (t_2 ^ 2.0))))));
	elseif (t_1 <= 5e+302)
		tmp = ((-2.0 * J) * cos((K * 0.5))) * sqrt((1.0 + ((0.5 * (U / J)) ^ 2.0)));
	else
		tmp = -2.0 * (-0.5 * U);
	end
	tmp_2 = 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[(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]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(t$95$2 * N[(-1.0 * N[(U * N[Sqrt[N[(0.25 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+302], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in U around -inf

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

   7. Applied rewrites 26.4%

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

   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))))) < 5e302

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 64.3%

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

   if 5e302 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in U around -inf

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

   10. Applied rewrites 27.0%

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

Alternative 3: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;2 \cdot \left(U \cdot \left(t\_2 \cdot \sqrt{\frac{0.25}{{t\_2}^{2}}}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0)))))
        (t_2 (cos (* 0.5 K))))
   (if (<= t_1 (- INFINITY))
     (* 2.0 (* U (* t_2 (sqrt (/ 0.25 (pow t_2 2.0))))))
     (if (<= t_1 5e+302)
       (*
        (* (* -2.0 J) (cos (* K 0.5)))
        (sqrt (+ 1.0 (pow (* 0.5 (/ U J)) 2.0))))
       (* -2.0 (* -0.5 U))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
	double t_2 = cos((0.5 * K));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 2.0 * (U * (t_2 * sqrt((0.25 / pow(t_2, 2.0)))));
	} else if (t_1 <= 5e+302) {
		tmp = ((-2.0 * J) * cos((K * 0.5))) * sqrt((1.0 + pow((0.5 * (U / J)), 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
	double t_2 = Math.cos((0.5 * K));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 2.0 * (U * (t_2 * Math.sqrt((0.25 / Math.pow(t_2, 2.0)))));
	} else if (t_1 <= 5e+302) {
		tmp = ((-2.0 * J) * Math.cos((K * 0.5))) * Math.sqrt((1.0 + Math.pow((0.5 * (U / J)), 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))
	t_2 = math.cos((0.5 * K))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 2.0 * (U * (t_2 * math.sqrt((0.25 / math.pow(t_2, 2.0)))))
	elif t_1 <= 5e+302:
		tmp = ((-2.0 * J) * math.cos((K * 0.5))) * math.sqrt((1.0 + math.pow((0.5 * (U / J)), 2.0)))
	else:
		tmp = -2.0 * (-0.5 * U)
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_2 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(2.0 * Float64(U * Float64(t_2 * sqrt(Float64(0.25 / (t_2 ^ 2.0))))));
	elseif (t_1 <= 5e+302)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U / J)) ^ 2.0))));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
	t_2 = cos((0.5 * K));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 2.0 * (U * (t_2 * sqrt((0.25 / (t_2 ^ 2.0)))));
	elseif (t_1 <= 5e+302)
		tmp = ((-2.0 * J) * cos((K * 0.5))) * sqrt((1.0 + ((0.5 * (U / J)) ^ 2.0)));
	else
		tmp = -2.0 * (-0.5 * U);
	end
	tmp_2 = 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[(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]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(2.0 * N[(U * N[(t$95$2 * N[Sqrt[N[(0.25 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+302], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.8%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in J around 0

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

   7. Applied rewrites 26.4%

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

   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))))) < 5e302

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 64.3%

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

   if 5e302 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in U around -inf

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

   10. Applied rewrites 27.0%

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

Alternative 4: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \left(-0.5 \cdot U\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* -0.5 U)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J) t_1)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_1)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     t_0
     (if (<= t_2 5e+302)
       (*
        (* (* -2.0 J) (cos (* K 0.5)))
        (sqrt (+ 1.0 (pow (* 0.5 (/ U J)) 2.0))))
       t_0))))

C

double code(double J, double K, double U) {
	double t_0 = -2.0 * (-0.5 * U);
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + pow((U / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= 5e+302) {
		tmp = ((-2.0 * J) * cos((K * 0.5))) * sqrt((1.0 + pow((0.5 * (U / J)), 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (-0.5 * U);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = ((-2.0 * J) * t_1) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_2 <= 5e+302) {
		tmp = ((-2.0 * J) * Math.cos((K * 0.5))) * Math.sqrt((1.0 + Math.pow((0.5 * (U / J)), 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = -2.0 * (-0.5 * U)
	t_1 = math.cos((K / 2.0))
	t_2 = ((-2.0 * J) * t_1) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_1)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_0
	elif t_2 <= 5e+302:
		tmp = ((-2.0 * J) * math.cos((K * 0.5))) * math.sqrt((1.0 + math.pow((0.5 * (U / J)), 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(-0.5 * U))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J) * t_1) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= 5e+302)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U / J)) ^ 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (-0.5 * U);
	t_1 = cos((K / 2.0));
	t_2 = ((-2.0 * J) * t_1) * sqrt((1.0 + ((U / ((2.0 * J) * t_1)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_0;
	elseif (t_2 <= 5e+302)
		tmp = ((-2.0 * J) * cos((K * 0.5))) * sqrt((1.0 + ((0.5 * (U / J)) ^ 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(-0.5 * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, 5e+302], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision], 2.0], $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 5e302 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in U around -inf

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

   10. Applied rewrites 27.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5e302

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 64.3%

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

Alternative 5: 58.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right) \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= K 8e-6)
   (*
    (fma -2.0 J (* 0.25 (* J (pow K 2.0))))
    (sqrt (+ 1.0 (pow (* 0.5 (/ U J)) 2.0))))
   (* (* (* -2.0 J) (cos (* K 0.5))) 1.0)))

C

double code(double J, double K, double U) {
	double tmp;
	if (K <= 8e-6) {
		tmp = fma(-2.0, J, (0.25 * (J * pow(K, 2.0)))) * sqrt((1.0 + pow((0.5 * (U / J)), 2.0)));
	} else {
		tmp = ((-2.0 * J) * cos((K * 0.5))) * 1.0;
	}
	return tmp;
}

Julia

function code(J, K, U)
	tmp = 0.0
	if (K <= 8e-6)
		tmp = Float64(fma(-2.0, J, Float64(0.25 * Float64(J * (K ^ 2.0)))) * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U / J)) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5))) * 1.0);
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := If[LessEqual[K, 8e-6], N[(N[(-2.0 * J + N[(0.25 * N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 7.99999999999999964e-6

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   4. Taylor expanded in K around 0

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

   6. Applied rewrites 64.3%

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

   7. Taylor expanded in K around 0

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

   9. Applied rewrites 38.2%

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

   if 7.99999999999999964e-6 < K

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   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 51.6%

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

Alternative 6: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \left(-0.5 \cdot U\right)\\ \mathbf{if}\;U \leq -3.3 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U \leq -4.3 \cdot 10^{-98}:\\ \;\;\;\;-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)\\ \mathbf{elif}\;U \leq 3.6 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* -0.5 U))))
   (if (<= U -3.3e+147)
     t_0
     (if (<= U -4.3e-98)
       (* -2.0 (* J (sqrt (+ 1.0 (* 0.25 (/ (pow U 2.0) (pow J 2.0)))))))
       (if (<= U 3.6e+139) (* (* (* -2.0 J) (cos (* K 0.5))) 1.0) t_0)))))

C

double code(double J, double K, double U) {
	double t_0 = -2.0 * (-0.5 * U);
	double tmp;
	if (U <= -3.3e+147) {
		tmp = t_0;
	} else if (U <= -4.3e-98) {
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * (pow(U, 2.0) / pow(J, 2.0))))));
	} else if (U <= 3.6e+139) {
		tmp = ((-2.0 * J) * cos((K * 0.5))) * 1.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) * ((-0.5d0) * u)
    if (u <= (-3.3d+147)) then
        tmp = t_0
    else if (u <= (-4.3d-98)) then
        tmp = (-2.0d0) * (j * sqrt((1.0d0 + (0.25d0 * ((u ** 2.0d0) / (j ** 2.0d0))))))
    else if (u <= 3.6d+139) then
        tmp = (((-2.0d0) * j) * cos((k * 0.5d0))) * 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (-0.5 * U);
	double tmp;
	if (U <= -3.3e+147) {
		tmp = t_0;
	} else if (U <= -4.3e-98) {
		tmp = -2.0 * (J * Math.sqrt((1.0 + (0.25 * (Math.pow(U, 2.0) / Math.pow(J, 2.0))))));
	} else if (U <= 3.6e+139) {
		tmp = ((-2.0 * J) * Math.cos((K * 0.5))) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = -2.0 * (-0.5 * U)
	tmp = 0
	if U <= -3.3e+147:
		tmp = t_0
	elif U <= -4.3e-98:
		tmp = -2.0 * (J * math.sqrt((1.0 + (0.25 * (math.pow(U, 2.0) / math.pow(J, 2.0))))))
	elif U <= 3.6e+139:
		tmp = ((-2.0 * J) * math.cos((K * 0.5))) * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(-0.5 * U))
	tmp = 0.0
	if (U <= -3.3e+147)
		tmp = t_0;
	elseif (U <= -4.3e-98)
		tmp = Float64(-2.0 * Float64(J * sqrt(Float64(1.0 + Float64(0.25 * Float64((U ^ 2.0) / (J ^ 2.0)))))));
	elseif (U <= 3.6e+139)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5))) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (-0.5 * U);
	tmp = 0.0;
	if (U <= -3.3e+147)
		tmp = t_0;
	elseif (U <= -4.3e-98)
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * ((U ^ 2.0) / (J ^ 2.0))))));
	elseif (U <= 3.6e+139)
		tmp = ((-2.0 * J) * cos((K * 0.5))) * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(-0.5 * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -3.3e+147], t$95$0, If[LessEqual[U, -4.3e-98], 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], If[LessEqual[U, 3.6e+139], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if U < -3.30000000000000025e147 or 3.59999999999999985e139 < U

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in U around -inf

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

   10. Applied rewrites 27.0%

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

   if -3.30000000000000025e147 < U < -4.29999999999999988e-98

   1. Initial program 72.8%

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

   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 33.2%

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

   if -4.29999999999999988e-98 < U < 3.59999999999999985e139

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   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 51.6%

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

Alternative 7: 50.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -8.8 \cdot 10^{+33}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2}}}\right)\right)\\ \mathbf{elif}\;U \leq 3.6 \cdot 10^{+139}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= U -8.8e+33)
   (* 2.0 (* J (* U (sqrt (/ 0.25 (pow J 2.0))))))
   (if (<= U 3.6e+139)
     (* (* (* -2.0 J) (cos (* K 0.5))) 1.0)
     (* -2.0 (* -0.5 U)))))

C

double code(double J, double K, double U) {
	double tmp;
	if (U <= -8.8e+33) {
		tmp = 2.0 * (J * (U * sqrt((0.25 / pow(J, 2.0)))));
	} else if (U <= 3.6e+139) {
		tmp = ((-2.0 * J) * cos((K * 0.5))) * 1.0;
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	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 (u <= (-8.8d+33)) then
        tmp = 2.0d0 * (j * (u * sqrt((0.25d0 / (j ** 2.0d0)))))
    else if (u <= 3.6d+139) then
        tmp = (((-2.0d0) * j) * cos((k * 0.5d0))) * 1.0d0
    else
        tmp = (-2.0d0) * ((-0.5d0) * u)
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (U <= -8.8e+33) {
		tmp = 2.0 * (J * (U * Math.sqrt((0.25 / Math.pow(J, 2.0)))));
	} else if (U <= 3.6e+139) {
		tmp = ((-2.0 * J) * Math.cos((K * 0.5))) * 1.0;
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if U <= -8.8e+33:
		tmp = 2.0 * (J * (U * math.sqrt((0.25 / math.pow(J, 2.0)))))
	elif U <= 3.6e+139:
		tmp = ((-2.0 * J) * math.cos((K * 0.5))) * 1.0
	else:
		tmp = -2.0 * (-0.5 * U)
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (U <= -8.8e+33)
		tmp = Float64(2.0 * Float64(J * Float64(U * sqrt(Float64(0.25 / (J ^ 2.0))))));
	elseif (U <= 3.6e+139)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5))) * 1.0);
	else
		tmp = Float64(-2.0 * Float64(-0.5 * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= -8.8e+33)
		tmp = 2.0 * (J * (U * sqrt((0.25 / (J ^ 2.0)))));
	elseif (U <= 3.6e+139)
		tmp = ((-2.0 * J) * cos((K * 0.5))) * 1.0;
	else
		tmp = -2.0 * (-0.5 * U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[U, -8.8e+33], N[(2.0 * N[(J * N[(U * N[Sqrt[N[(0.25 / N[Power[J, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 3.6e+139], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(-2.0 * N[(-0.5 * U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -8.79999999999999975e33

   1. Initial program 72.8%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 13.1%

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

   if -8.79999999999999975e33 < U < 3.59999999999999985e139

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   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 51.6%

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

   if 3.59999999999999985e139 < U

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in U around -inf

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

   10. Applied rewrites 27.0%

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

Alternative 8: 38.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -1.06 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(U \cdot \sqrt{\frac{0.25}{{J}^{2}}}\right)\right)\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= U -1.06e+30)
   (* 2.0 (* J (* U (sqrt (/ 0.25 (pow J 2.0))))))
   (if (<= U 2.4e+75)
     (* (fma -2.0 J (* 0.25 (* J (pow K 2.0)))) 1.0)
     (* -2.0 (* -0.5 U)))))

C

double code(double J, double K, double U) {
	double tmp;
	if (U <= -1.06e+30) {
		tmp = 2.0 * (J * (U * sqrt((0.25 / pow(J, 2.0)))));
	} else if (U <= 2.4e+75) {
		tmp = fma(-2.0, J, (0.25 * (J * pow(K, 2.0)))) * 1.0;
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	return tmp;
}

Julia

function code(J, K, U)
	tmp = 0.0
	if (U <= -1.06e+30)
		tmp = Float64(2.0 * Float64(J * Float64(U * sqrt(Float64(0.25 / (J ^ 2.0))))));
	elseif (U <= 2.4e+75)
		tmp = Float64(fma(-2.0, J, Float64(0.25 * Float64(J * (K ^ 2.0)))) * 1.0);
	else
		tmp = Float64(-2.0 * Float64(-0.5 * U));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := If[LessEqual[U, -1.06e+30], N[(2.0 * N[(J * N[(U * N[Sqrt[N[(0.25 / N[Power[J, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 2.4e+75], N[(N[(-2.0 * J + N[(0.25 * N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(-2.0 * N[(-0.5 * U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if U < -1.06e30

   1. Initial program 72.8%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 13.1%

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

   if -1.06e30 < U < 2.4e75

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   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 51.6%

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

   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 27.3%

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

   if 2.4e75 < U

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in U around -inf

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

   10. Applied rewrites 27.0%

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

Alternative 9: 36.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \left(-0.5 \cdot U\right)\\ \mathbf{if}\;U \leq -3.8 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* -0.5 U))))
   (if (<= U -3.8e-26)
     t_0
     (if (<= U 2.4e+75) (* (fma -2.0 J (* 0.25 (* J (pow K 2.0)))) 1.0) t_0))))

C

double code(double J, double K, double U) {
	double t_0 = -2.0 * (-0.5 * U);
	double tmp;
	if (U <= -3.8e-26) {
		tmp = t_0;
	} else if (U <= 2.4e+75) {
		tmp = fma(-2.0, J, (0.25 * (J * pow(K, 2.0)))) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(-0.5 * U))
	tmp = 0.0
	if (U <= -3.8e-26)
		tmp = t_0;
	elseif (U <= 2.4e+75)
		tmp = Float64(fma(-2.0, J, Float64(0.25 * Float64(J * (K ^ 2.0)))) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(-0.5 * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -3.8e-26], t$95$0, If[LessEqual[U, 2.4e+75], N[(N[(-2.0 * J + N[(0.25 * N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if U < -3.80000000000000015e-26 or 2.4e75 < U

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in U around -inf

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

   10. Applied rewrites 27.0%

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

   if -3.80000000000000015e-26 < U < 2.4e75

   1. Initial program 72.8%

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

   3. Applied rewrites 72.8%

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

   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 51.6%

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

   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 27.3%

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

Alternative 10: 29.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \leq -1 \cdot 10^{-138}:\\ \;\;\;\;-2 \cdot \sqrt{0.25 \cdot {U}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<=
        (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))
        -1e-138)
     (* -2.0 (sqrt (* 0.25 (pow U 2.0))))
     (* -2.0 (* -0.5 U)))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)))) <= -1e-138) {
		tmp = -2.0 * sqrt((0.25 * pow(U, 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))) <= (-1d-138)) then
        tmp = (-2.0d0) * sqrt((0.25d0 * (u ** 2.0d0)))
    else
        tmp = (-2.0d0) * ((-0.5d0) * u)
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)))) <= -1e-138) {
		tmp = -2.0 * Math.sqrt((0.25 * Math.pow(U, 2.0)));
	} else {
		tmp = -2.0 * (-0.5 * U);
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))) <= -1e-138:
		tmp = -2.0 * math.sqrt((0.25 * math.pow(U, 2.0)))
	else:
		tmp = -2.0 * (-0.5 * U)
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0)))) <= -1e-138)
		tmp = Float64(-2.0 * sqrt(Float64(0.25 * (U ^ 2.0))));
	else
		tmp = Float64(-2.0 * Float64(-0.5 * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)))) <= -1e-138)
		tmp = -2.0 * sqrt((0.25 * (U ^ 2.0)));
	else
		tmp = -2.0 * (-0.5 * U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -1e-138], N[(-2.0 * N[Sqrt[N[(0.25 * N[Power[U, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(-0.5 * U), $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))))) < -1.00000000000000007e-138

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   if -1.00000000000000007e-138 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 72.8%

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

   2. Taylor expanded in J around 0

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

   4. Applied rewrites 15.5%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 16.1%

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

   8. Taylor expanded in U around -inf

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

   10. Applied rewrites 27.0%

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

Alternative 11: 27.0% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

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) * ((-0.5d0) * u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

2. Taylor expanded in J around 0

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

4. Applied rewrites 15.5%

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

5. Taylor expanded in K around 0

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

7. Applied rewrites 16.1%

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

8. Taylor expanded in U around -inf

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

10. Applied rewrites 27.0%

    \[\leadsto -2 \cdot \left(-0.5 \cdot U\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025160 
(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)))))