Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.6% → 99.1%

Time: 16.7s

Alternatives: 12

Speedup: 1.9×

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

real(8) function code(j, k, u)
    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: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;t_1 \leq 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(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 (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (* -2.0 (* U 0.5))
     (if (<= t_1 1e+300) t_1 (* -2.0 (* U -0.5))))))

C

U = abs(U);
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 / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -2.0 * (U * 0.5);
	} else if (t_1 <= 1e+300) {
		tmp = t_1;
	} else {
		tmp = -2.0 * (U * -0.5);
	}
	return tmp;
}

Java

U = Math.abs(U);
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 / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * (U * 0.5);
	} else if (t_1 <= 1e+300) {
		tmp = t_1;
	} else {
		tmp = -2.0 * (U * -0.5);
	}
	return tmp;
}

Python

U = abs(U)
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 / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -2.0 * (U * 0.5)
	elif t_1 <= 1e+300:
		tmp = t_1
	else:
		tmp = -2.0 * (U * -0.5)
	return tmp

Julia

U = abs(U)
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(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(U * 0.5));
	elseif (t_1 <= 1e+300)
		tmp = t_1;
	else
		tmp = Float64(-2.0 * Float64(U * -0.5));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -2.0 * (U * 0.5);
	elseif (t_1 <= 1e+300)
		tmp = t_1;
	else
		tmp = -2.0 * (U * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
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[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+300], t$95$1, N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.2%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in J around 0 36.1%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 1.0000000000000001e300

   1. Initial program 99.8%

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

   if 1.0000000000000001e300 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 8.0%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in U around -inf 53.2%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 53.2%

         \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5\right)} \]

   6. Simplified 53.2%

      \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+300}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \]

Alternative 2: 78.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{if}\;J \leq -9 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U, \frac{{\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (* (cos (/ K 2.0)) (hypot 1.0 (* U (/ 0.5 J))))))))
   (if (<= J -9e-106)
     t_0
     (if (<= J -2e-310)
       (* -2.0 (- (* U -0.5) (/ (* J J) U)))
       (if (<= J 1.2e-94)
         (* -2.0 (fma 0.5 U (/ (pow (* J (cos (* K 0.5))) 2.0) U)))
         t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (cos((K / 2.0)) * hypot(1.0, (U * (0.5 / J)))));
	double tmp;
	if (J <= -9e-106) {
		tmp = t_0;
	} else if (J <= -2e-310) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 1.2e-94) {
		tmp = -2.0 * fma(0.5, U, (pow((J * cos((K * 0.5))), 2.0) / U));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * Float64(cos(Float64(K / 2.0)) * hypot(1.0, Float64(U * Float64(0.5 / J))))))
	tmp = 0.0
	if (J <= -9e-106)
		tmp = t_0;
	elseif (J <= -2e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(Float64(J * J) / U)));
	elseif (J <= 1.2e-94)
		tmp = Float64(-2.0 * fma(0.5, U, Float64((Float64(J * cos(Float64(K * 0.5))) ^ 2.0) / U)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -9e-106], t$95$0, If[LessEqual[J, -2e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(N[(J * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.2e-94], N[(-2.0 * N[(0.5 * U + N[(N[Power[N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -8.99999999999999911e-106 or 1.2e-94 < J

   1. Initial program 87.6%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in K around 0 85.1%

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

      1. associate-*r/ 85.1%

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

      2. *-commutative 85.1%

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

      3. associate-*r/ 85.1%

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

   6. Simplified 85.1%

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

   if -8.99999999999999911e-106 < J < -1.999999999999994e-310

   1. Initial program 42.4%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in K around 0 12.3%

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

      1. unpow2 12.3%

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

      2. unpow2 12.3%

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

   6. Simplified 12.3%

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

   7. Taylor expanded in U around -inf 51.6%

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

      1. +-commutative 51.6%

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

      2. mul-1-neg 51.6%

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

      3. unsub-neg 51.6%

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

      4. *-commutative 51.6%

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

      5. unpow2 51.6%

         \[\leadsto -2 \cdot \left(U \cdot -0.5 - \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 51.6%

      \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)} \]

   if -1.999999999999994e-310 < J < 1.2e-94

   1. Initial program 41.5%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in J around 0 46.2%

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

      1. fma-def 46.2%

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

      2. unpow2 46.2%

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

      3. *-commutative 46.2%

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

      4. unpow2 46.2%

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

      5. swap-sqr 46.2%

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

      6. unpow2 46.2%

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

      7. *-commutative 46.2%

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

   6. Simplified 46.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9 \cdot 10^{-106}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(0.5, U, \frac{{\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \end{array} \]

Alternative 3: 88.0% accurate, 1.3× speedup

Math

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

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* -2.0 (* J (* t_0 (hypot 1.0 (/ U (* J (* 2.0 t_0)))))))))

C

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

Java

U = Math.abs(U);
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.hypot(1.0, (U / (J * (2.0 * t_0))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(-2.0 * N[(J * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.5%

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

3. Simplified 86.3%

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

4. Final simplification 86.3%

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

Alternative 4: 88.0% accurate, 1.3× speedup

Math

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

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* -2.0 (* t_0 (* J (hypot 1.0 (* 0.5 (/ U (* J t_0)))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. associate-*l* 71.5%

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

   2. associate-*l* 71.5%

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

   3. *-commutative 71.5%

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

   4. unpow2 71.5%

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

   5. sqr-neg 71.5%

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

   6. distribute-frac-neg 71.5%

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

   7. distribute-frac-neg 71.5%

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

   8. unpow2 71.5%

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

3. Simplified 86.3%

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

4. Final simplification 86.3%

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

Alternative 5: 88.1% accurate, 1.3× speedup

Math

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

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (cos (/ K 2.0)))))
   (* -2.0 (* t_0 (hypot 1.0 (/ U (* 2.0 t_0)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(-2.0 * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.5%

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

   1. associate-*l* 71.5%

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

   2. associate-*l* 71.5%

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

   3. unpow2 71.5%

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

   4. sqr-neg 71.5%

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

   5. distribute-frac-neg 71.5%

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

   6. distribute-frac-neg 71.5%

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

   7. unpow2 71.5%

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

3. Simplified 86.4%

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

4. Final simplification 86.4%

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

Alternative 6: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{if}\;J \leq -6.5 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-97}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (* (cos (/ K 2.0)) (hypot 1.0 (* U (/ 0.5 J))))))))
   (if (<= J -6.5e-104)
     t_0
     (if (<= J -2e-310)
       (* -2.0 (- (* U -0.5) (/ (* J J) U)))
       (if (<= J 2.6e-97) (* -2.0 (* U 0.5)) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (cos((K / 2.0)) * hypot(1.0, (U * (0.5 / J)))));
	double tmp;
	if (J <= -6.5e-104) {
		tmp = t_0;
	} else if (J <= -2e-310) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 2.6e-97) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (Math.cos((K / 2.0)) * Math.hypot(1.0, (U * (0.5 / J)))));
	double tmp;
	if (J <= -6.5e-104) {
		tmp = t_0;
	} else if (J <= -2e-310) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 2.6e-97) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J * (math.cos((K / 2.0)) * math.hypot(1.0, (U * (0.5 / J)))))
	tmp = 0
	if J <= -6.5e-104:
		tmp = t_0
	elif J <= -2e-310:
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U))
	elif J <= 2.6e-97:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * Float64(cos(Float64(K / 2.0)) * hypot(1.0, Float64(U * Float64(0.5 / J))))))
	tmp = 0.0
	if (J <= -6.5e-104)
		tmp = t_0;
	elseif (J <= -2e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(Float64(J * J) / U)));
	elseif (J <= 2.6e-97)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J * (cos((K / 2.0)) * hypot(1.0, (U * (0.5 / J)))));
	tmp = 0.0;
	if (J <= -6.5e-104)
		tmp = t_0;
	elseif (J <= -2e-310)
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	elseif (J <= 2.6e-97)
		tmp = -2.0 * (U * 0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U * N[(0.5 / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -6.5e-104], t$95$0, If[LessEqual[J, -2e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(N[(J * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 2.6e-97], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -6.49999999999999991e-104 or 2.60000000000000007e-97 < J

   1. Initial program 87.6%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in K around 0 85.1%

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

      1. associate-*r/ 85.1%

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

      2. *-commutative 85.1%

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

      3. associate-*r/ 85.1%

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

   6. Simplified 85.1%

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

   if -6.49999999999999991e-104 < J < -1.999999999999994e-310

   1. Initial program 42.4%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in K around 0 12.3%

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

      1. unpow2 12.3%

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

      2. unpow2 12.3%

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

   6. Simplified 12.3%

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

   7. Taylor expanded in U around -inf 51.6%

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

      1. +-commutative 51.6%

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

      2. mul-1-neg 51.6%

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

      3. unsub-neg 51.6%

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

      4. *-commutative 51.6%

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

      5. unpow2 51.6%

         \[\leadsto -2 \cdot \left(U \cdot -0.5 - \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 51.6%

      \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)} \]

   if -1.999999999999994e-310 < J < 2.60000000000000007e-97

   1. Initial program 41.5%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in J around 0 46.3%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -6.5 \cdot 10^{-104}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-97}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\right)\\ \end{array} \]

Alternative 7: 61.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{+27}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{elif}\;\frac{K}{2} \leq 5 \cdot 10^{+222} \lor \neg \left(\frac{K}{2} \leq 5 \cdot 10^{+241}\right):\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e+27)
   (* -2.0 (* J (hypot 1.0 (/ (* U 0.5) J))))
   (if (or (<= (/ K 2.0) 5e+222) (not (<= (/ K 2.0) 5e+241)))
     (* -2.0 (* J (cos (/ K 2.0))))
     (* -2.0 (- (* U -0.5) (/ (* J J) U))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e+27) {
		tmp = -2.0 * (J * hypot(1.0, ((U * 0.5) / J)));
	} else if (((K / 2.0) <= 5e+222) || !((K / 2.0) <= 5e+241)) {
		tmp = -2.0 * (J * cos((K / 2.0)));
	} else {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e+27) {
		tmp = -2.0 * (J * Math.hypot(1.0, ((U * 0.5) / J)));
	} else if (((K / 2.0) <= 5e+222) || !((K / 2.0) <= 5e+241)) {
		tmp = -2.0 * (J * Math.cos((K / 2.0)));
	} else {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if (K / 2.0) <= 5e+27:
		tmp = -2.0 * (J * math.hypot(1.0, ((U * 0.5) / J)))
	elif ((K / 2.0) <= 5e+222) or not ((K / 2.0) <= 5e+241):
		tmp = -2.0 * (J * math.cos((K / 2.0)))
	else:
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e+27)
		tmp = Float64(-2.0 * Float64(J * hypot(1.0, Float64(Float64(U * 0.5) / J))));
	elseif ((Float64(K / 2.0) <= 5e+222) || !(Float64(K / 2.0) <= 5e+241))
		tmp = Float64(-2.0 * Float64(J * cos(Float64(K / 2.0))));
	else
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(Float64(J * J) / U)));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e+27)
		tmp = -2.0 * (J * hypot(1.0, ((U * 0.5) / J)));
	elseif (((K / 2.0) <= 5e+222) || ~(((K / 2.0) <= 5e+241)))
		tmp = -2.0 * (J * cos((K / 2.0)));
	else
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e+27], N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(K / 2.0), $MachinePrecision], 5e+222], N[Not[LessEqual[N[(K / 2.0), $MachinePrecision], 5e+241]], $MachinePrecision]], N[(-2.0 * N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(N[(J * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 K 2) < 4.99999999999999979e27

   1. Initial program 73.1%

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

   3. Simplified 86.8%

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

      1. add-cube-cbrt 85.8%

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}\right)}\right) \]

      2. pow3 85.8%

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

   5. Applied egg-rr 85.9%

      \[\leadsto -2 \cdot \left(J \cdot \color{blue}{{\left(\sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{0.5 \cdot \frac{U}{J}}{\cos \left(K \cdot 0.5\right)}\right)}\right)}^{3}}\right) \]

   6. Taylor expanded in K around 0 40.5%

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

      1. metadata-eval 40.5%

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

      2. unpow2 40.5%

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

      3. unpow2 40.5%

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

      4. times-frac 51.9%

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

      5. swap-sqr 52.4%

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

      6. unpow2 52.4%

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

      7. associate-*r/ 52.4%

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

      8. *-commutative 52.4%

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

      9. *-commutative 52.4%

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

      10. associate-*r/ 52.4%

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

      11. unpow2 52.4%

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

      12. hypot-1-def 64.7%

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

      13. associate-*r/ 64.7%

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

      14. *-commutative 64.7%

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

   8. Simplified 64.7%

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

   if 4.99999999999999979e27 < (/.f64 K 2) < 5.00000000000000023e222 or 5.00000000000000025e241 < (/.f64 K 2)

   1. Initial program 67.2%

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

      1. associate-*l* 67.2%

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

      2. associate-*l* 67.2%

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

      3. *-commutative 67.2%

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

      4. unpow2 67.2%

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

      5. sqr-neg 67.2%

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

      6. distribute-frac-neg 67.2%

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

      7. distribute-frac-neg 67.2%

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

      8. unpow2 67.2%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in J around inf 42.9%

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

   if 5.00000000000000023e222 < (/.f64 K 2) < 5.00000000000000025e241

   1. Initial program 6.3%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in K around 0 0.3%

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

      1. unpow2 0.3%

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

      2. unpow2 0.3%

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

   6. Simplified 0.3%

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

   7. Taylor expanded in U around -inf 50.0%

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

      1. +-commutative 50.0%

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

      2. mul-1-neg 50.0%

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

      3. unsub-neg 50.0%

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

      4. *-commutative 50.0%

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

      5. unpow2 50.0%

         \[\leadsto -2 \cdot \left(U \cdot -0.5 - \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 50.0%

      \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{+27}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{elif}\;\frac{K}{2} \leq 5 \cdot 10^{+222} \lor \neg \left(\frac{K}{2} \leq 5 \cdot 10^{+241}\right):\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \end{array} \]

Alternative 8: 67.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{if}\;J \leq -4.4 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 2.25 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (cos (/ K 2.0))))))
   (if (<= J -4.4e+42)
     t_0
     (if (<= J -2e-310)
       (* -2.0 (- (* U -0.5) (/ (* J J) U)))
       (if (<= J 2.25e-38) (* -2.0 (* U 0.5)) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * cos((K / 2.0)));
	double tmp;
	if (J <= -4.4e+42) {
		tmp = t_0;
	} else if (J <= -2e-310) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 2.25e-38) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    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) * (j * cos((k / 2.0d0)))
    if (j <= (-4.4d+42)) then
        tmp = t_0
    else if (j <= (-2d-310)) then
        tmp = (-2.0d0) * ((u * (-0.5d0)) - ((j * j) / u))
    else if (j <= 2.25d-38) then
        tmp = (-2.0d0) * (u * 0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * Math.cos((K / 2.0)));
	double tmp;
	if (J <= -4.4e+42) {
		tmp = t_0;
	} else if (J <= -2e-310) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 2.25e-38) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J * math.cos((K / 2.0)))
	tmp = 0
	if J <= -4.4e+42:
		tmp = t_0
	elif J <= -2e-310:
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U))
	elif J <= 2.25e-38:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * cos(Float64(K / 2.0))))
	tmp = 0.0
	if (J <= -4.4e+42)
		tmp = t_0;
	elseif (J <= -2e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(Float64(J * J) / U)));
	elseif (J <= 2.25e-38)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J * cos((K / 2.0)));
	tmp = 0.0;
	if (J <= -4.4e+42)
		tmp = t_0;
	elseif (J <= -2e-310)
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	elseif (J <= 2.25e-38)
		tmp = -2.0 * (U * 0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -4.4e+42], t$95$0, If[LessEqual[J, -2e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(N[(J * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 2.25e-38], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -4.4000000000000003e42 or 2.25000000000000004e-38 < J

   1. Initial program 94.5%

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

      1. associate-*l* 94.5%

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

      2. associate-*l* 94.5%

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

      3. *-commutative 94.5%

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

      4. unpow2 94.5%

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

      5. sqr-neg 94.5%

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

      6. distribute-frac-neg 94.5%

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

      7. distribute-frac-neg 94.5%

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

      8. unpow2 94.5%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in J around inf 76.5%

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

   if -4.4000000000000003e42 < J < -1.999999999999994e-310

   1. Initial program 54.2%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in K around 0 26.0%

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

      1. unpow2 26.0%

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

      2. unpow2 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in U around -inf 44.0%

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

      1. +-commutative 44.0%

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

      2. mul-1-neg 44.0%

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

      3. unsub-neg 44.0%

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

      4. *-commutative 44.0%

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

      5. unpow2 44.0%

         \[\leadsto -2 \cdot \left(U \cdot -0.5 - \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 44.0%

      \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)} \]

   if -1.999999999999994e-310 < J < 2.25000000000000004e-38

   1. Initial program 47.0%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in J around 0 43.1%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.4 \cdot 10^{+42}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 2.25 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array} \]

Alternative 9: 50.0% accurate, 27.8× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -7 \cdot 10^{+41}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 1.4 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -7e+41)
   (* -2.0 J)
   (if (<= J -2e-310)
     (* -2.0 (- (* U -0.5) (/ (* J J) U)))
     (if (<= J 1.4e-37) (* -2.0 (* U 0.5)) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -7e+41) {
		tmp = -2.0 * J;
	} else if (J <= -2e-310) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 1.4e-37) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-7d+41)) then
        tmp = (-2.0d0) * j
    else if (j <= (-2d-310)) then
        tmp = (-2.0d0) * ((u * (-0.5d0)) - ((j * j) / u))
    else if (j <= 1.4d-37) then
        tmp = (-2.0d0) * (u * 0.5d0)
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -7e+41) {
		tmp = -2.0 * J;
	} else if (J <= -2e-310) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 1.4e-37) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -7e+41:
		tmp = -2.0 * J
	elif J <= -2e-310:
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U))
	elif J <= 1.4e-37:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -7e+41)
		tmp = Float64(-2.0 * J);
	elseif (J <= -2e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(Float64(J * J) / U)));
	elseif (J <= 1.4e-37)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -7e+41)
		tmp = -2.0 * J;
	elseif (J <= -2e-310)
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	elseif (J <= 1.4e-37)
		tmp = -2.0 * (U * 0.5);
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -7e+41], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -2e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(N[(J * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.4e-37], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -6.9999999999999998e41 or 1.4000000000000001e-37 < J

   1. Initial program 94.5%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in K around 0 47.6%

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

      1. unpow2 47.6%

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

      2. unpow2 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in J around inf 47.0%

      \[\leadsto -2 \cdot \color{blue}{J} \]

   if -6.9999999999999998e41 < J < -1.999999999999994e-310

   1. Initial program 54.2%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in K around 0 26.0%

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

      1. unpow2 26.0%

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

      2. unpow2 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in U around -inf 44.0%

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

      1. +-commutative 44.0%

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

      2. mul-1-neg 44.0%

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

      3. unsub-neg 44.0%

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

      4. *-commutative 44.0%

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

      5. unpow2 44.0%

         \[\leadsto -2 \cdot \left(U \cdot -0.5 - \frac{\color{blue}{J \cdot J}}{U}\right) \]

   9. Simplified 44.0%

      \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)} \]

   if -1.999999999999994e-310 < J < 1.4000000000000001e-37

   1. Initial program 47.0%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in J around 0 43.1%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -7 \cdot 10^{+41}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 1.4 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 10: 48.3% accurate, 37.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -9.2 \cdot 10^{+184}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -9.2e+184)
   (* -2.0 J)
   (if (<= J -2e-310)
     (* -2.0 (* U -0.5))
     (if (<= J 1.45e-37) (* -2.0 (* U 0.5)) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -9.2e+184) {
		tmp = -2.0 * J;
	} else if (J <= -2e-310) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.45e-37) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-9.2d+184)) then
        tmp = (-2.0d0) * j
    else if (j <= (-2d-310)) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    else if (j <= 1.45d-37) then
        tmp = (-2.0d0) * (u * 0.5d0)
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -9.2e+184) {
		tmp = -2.0 * J;
	} else if (J <= -2e-310) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 1.45e-37) {
		tmp = -2.0 * (U * 0.5);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -9.2e+184:
		tmp = -2.0 * J
	elif J <= -2e-310:
		tmp = -2.0 * (U * -0.5)
	elif J <= 1.45e-37:
		tmp = -2.0 * (U * 0.5)
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -9.2e+184)
		tmp = Float64(-2.0 * J);
	elseif (J <= -2e-310)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 1.45e-37)
		tmp = Float64(-2.0 * Float64(U * 0.5));
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -9.2e+184)
		tmp = -2.0 * J;
	elseif (J <= -2e-310)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 1.45e-37)
		tmp = -2.0 * (U * 0.5);
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -9.2e+184], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -2e-310], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 1.45e-37], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -9.1999999999999999e184 or 1.45000000000000002e-37 < J

   1. Initial program 96.1%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in K around 0 50.2%

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

      1. unpow2 50.2%

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

      2. unpow2 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in J around inf 50.5%

      \[\leadsto -2 \cdot \color{blue}{J} \]

   if -9.1999999999999999e184 < J < -1.999999999999994e-310

   1. Initial program 61.7%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in U around -inf 39.4%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 39.4%

         \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5\right)} \]

   6. Simplified 39.4%

      \[\leadsto -2 \cdot \color{blue}{\left(U \cdot -0.5\right)} \]

   if -1.999999999999994e-310 < J < 1.45000000000000002e-37

   1. Initial program 47.0%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in J around 0 43.1%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9.2 \cdot 10^{+184}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 11: 40.4% accurate, 59.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 1.6 \cdot 10^{+25}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 1.6e+25) (* -2.0 J) (* -2.0 (* U 0.5))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.6e+25) {
		tmp = -2.0 * J;
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (u <= 1.6d+25) then
        tmp = (-2.0d0) * j
    else
        tmp = (-2.0d0) * (u * 0.5d0)
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.6e+25) {
		tmp = -2.0 * J;
	} else {
		tmp = -2.0 * (U * 0.5);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 1.6e+25:
		tmp = -2.0 * J
	else:
		tmp = -2.0 * (U * 0.5)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 1.6e+25)
		tmp = Float64(-2.0 * J);
	else
		tmp = Float64(-2.0 * Float64(U * 0.5));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 1.6e+25)
		tmp = -2.0 * J;
	else
		tmp = -2.0 * (U * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 1.6e+25], N[(-2.0 * J), $MachinePrecision], N[(-2.0 * N[(U * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 1.6e25

   1. Initial program 76.9%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in K around 0 40.1%

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

      1. unpow2 40.1%

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

      2. unpow2 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in J around inf 33.6%

      \[\leadsto -2 \cdot \color{blue}{J} \]

   if 1.6e25 < U

   1. Initial program 49.4%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in J around 0 41.1%

      \[\leadsto -2 \cdot \color{blue}{\left(0.5 \cdot U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.6 \cdot 10^{+25}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 29.0% accurate, 140.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ -2 \cdot J \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (* -2.0 J))

C

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

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = (-2.0d0) * j
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := N[(-2.0 * J), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

3. Simplified 86.3%

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

4. Taylor expanded in K around 0 35.0%

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

   1. unpow2 35.0%

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

   2. unpow2 35.0%

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

6. Simplified 35.0%

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

7. Taylor expanded in J around inf 28.8%

   \[\leadsto -2 \cdot \color{blue}{J} \]

8. Final simplification 28.8%

   \[\leadsto -2 \cdot J \]

Reproduce

herbie shell --seed 2023272 
(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)))))