Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.7% → 99.6%

Time: 8.7s

Alternatives: 9

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

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: 72.7% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1}\\ t_2 := t\_1 \cdot \left(\left(J \cdot -2\right) \cdot t\_0\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0)))
        (t_2 (* t_1 (* (* J -2.0) t_0))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 2e+305)
       (* (* (* (cos (* -0.5 K)) J) -2.0) t_1)
       (* -1.0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0));
	double t_2 = t_1 * ((J * -2.0) * t_0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 2e+305) {
		tmp = ((cos((-0.5 * K)) * J) * -2.0) * t_1;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.sqrt((Math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0));
	double t_2 = t_1 * ((J * -2.0) * t_0);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 2e+305) {
		tmp = ((Math.cos((-0.5 * K)) * J) * -2.0) * t_1;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.sqrt((math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0))
	t_2 = t_1 * ((J * -2.0) * t_0)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 2e+305:
		tmp = ((math.cos((-0.5 * K)) * J) * -2.0) * t_1
	else:
		tmp = -1.0 * -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0))
	t_2 = Float64(t_1 * Float64(Float64(J * -2.0) * t_0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 2e+305)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J) * -2.0) * t_1);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = sqrt((((U_m / ((2.0 * J) * t_0)) ^ 2.0) + 1.0));
	t_2 = t_1 * ((J * -2.0) * t_0);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 2e+305)
		tmp = ((cos((-0.5 * K)) * J) * -2.0) * t_1;
	else
		tmp = -1.0 * -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(J * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 2e+305], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(-1.0 * (-U$95$m)), $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 5.4%

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

   3. Taylor expanded in U around inf

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 51.7

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. div-inv N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.9999999999999999e305 < (*.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 5.3%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in U around inf

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 55.1%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_1}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot t\_1\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J}, \frac{U\_m}{J}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (* (cos (* 0.5 K)) (* J -2.0)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_1)) 2.0) 1.0))
          (* (* J -2.0) t_1))))
   (if (<= t_2 -5e+307)
     (- U_m)
     (if (<= t_2 -2e+190)
       t_0
       (if (<= t_2 -2e-153)
         (* (* J -2.0) (sqrt (fma (/ (* 0.25 U_m) J) (/ U_m J) 1.0)))
         (if (<= t_2 2e+305) t_0 (* -1.0 (- U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((0.5 * K)) * (J * -2.0);
	double t_1 = cos((K / 2.0));
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_1)), 2.0) + 1.0)) * ((J * -2.0) * t_1);
	double tmp;
	if (t_2 <= -5e+307) {
		tmp = -U_m;
	} else if (t_2 <= -2e+190) {
		tmp = t_0;
	} else if (t_2 <= -2e-153) {
		tmp = (J * -2.0) * sqrt(fma(((0.25 * U_m) / J), (U_m / J), 1.0));
	} else if (t_2 <= 2e+305) {
		tmp = t_0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = Float64(cos(Float64(0.5 * K)) * Float64(J * -2.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_1)) ^ 2.0) + 1.0)) * Float64(Float64(J * -2.0) * t_1))
	tmp = 0.0
	if (t_2 <= -5e+307)
		tmp = Float64(-U_m);
	elseif (t_2 <= -2e+190)
		tmp = t_0;
	elseif (t_2 <= -2e-153)
		tmp = Float64(Float64(J * -2.0) * sqrt(fma(Float64(Float64(0.25 * U_m) / J), Float64(U_m / J), 1.0)));
	elseif (t_2 <= 2e+305)
		tmp = t_0;
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J * -2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+307], (-U$95$m), If[LessEqual[t$95$2, -2e+190], t$95$0, If[LessEqual[t$95$2, -2e-153], N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+305], t$95$0, N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 8.1%

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

   3. Taylor expanded in U around inf

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 53.0

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{-U} \]

   if -5e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e190 or -2.00000000000000008e-153 < (*.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))))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

   3. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if -2.0000000000000001e190 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000008e-153

   1. Initial program 99.8%

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

   3. 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. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if 1.9999999999999999e305 < (*.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 5.3%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in U around inf

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 55.1%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -2 \cdot 10^{+190}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot t\_0\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, {\left(\cos \left(0.5 \cdot K\right) \cdot \frac{J}{U\_m}\right)}^{-2}, 1\right)} \cdot \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0))
          (* (* J -2.0) t_0))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 2e+305)
       (*
        (sqrt (fma 0.25 (pow (* (cos (* 0.5 K)) (/ J U_m)) -2.0) 1.0))
        (* (* (cos (* -0.5 K)) J) -2.0))
       (* -1.0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * ((J * -2.0) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e+305) {
		tmp = sqrt(fma(0.25, pow((cos((0.5 * K)) * (J / U_m)), -2.0), 1.0)) * ((cos((-0.5 * K)) * J) * -2.0);
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J * -2.0) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e+305)
		tmp = Float64(sqrt(fma(0.25, (Float64(cos(Float64(0.5 * K)) * Float64(J / U_m)) ^ -2.0), 1.0)) * Float64(Float64(cos(Float64(-0.5 * K)) * J) * -2.0));
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 2e+305], N[(N[Sqrt[N[(0.25 * N[Power[N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J / U$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 * (-U$95$m)), $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 5.4%

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

   3. Taylor expanded in U around inf

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 51.7

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. div-inv N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.8

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

   6. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. unpow-prod-down N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

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

   if 1.9999999999999999e305 < (*.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 5.3%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in U around inf

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 55.1%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, {\left(\cos \left(0.5 \cdot K\right) \cdot \frac{J}{U}\right)}^{-2}, 1\right)} \cdot \left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(J \cdot -2\right) \cdot t\_0\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J}, \frac{U\_m}{J}, 1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* J -2.0) t_0))
        (t_2 (* (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0)) t_1)))
   (if (<= t_2 -5e+307)
     (- U_m)
     (if (<= t_2 2e+305)
       (* (sqrt (fma (/ (* 0.25 U_m) J) (/ U_m J) 1.0)) t_1)
       (* -1.0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (J * -2.0) * t_0;
	double t_2 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * t_1;
	double tmp;
	if (t_2 <= -5e+307) {
		tmp = -U_m;
	} else if (t_2 <= 2e+305) {
		tmp = sqrt(fma(((0.25 * U_m) / J), (U_m / J), 1.0)) * t_1;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(J * -2.0) * t_0)
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * t_1)
	tmp = 0.0
	if (t_2 <= -5e+307)
		tmp = Float64(-U_m);
	elseif (t_2 <= 2e+305)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J), Float64(U_m / J), 1.0)) * t_1);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+307], (-U$95$m), If[LessEqual[t$95$2, 2e+305], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(-1.0 * (-U$95$m)), $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))))) < -5e307

   1. Initial program 8.1%

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

   3. Taylor expanded in U around inf

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 53.0

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{-U} \]

   if -5e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if 1.9999999999999999e305 < (*.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 5.3%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in U around inf

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 55.1%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot t\_0\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-246}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J}, \frac{U\_m}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot J, \frac{2}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0))
          (* (* J -2.0) t_0))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -5e-246)
       (* (* J -2.0) (sqrt (fma (/ (* 0.25 U_m) J) (/ U_m J) 1.0)))
       (fma (* J J) (/ 2.0 U_m) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * ((J * -2.0) * t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e-246) {
		tmp = (J * -2.0) * sqrt(fma(((0.25 * U_m) / J), (U_m / J), 1.0));
	} else {
		tmp = fma((J * J), (2.0 / U_m), U_m);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J * -2.0) * t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-246)
		tmp = Float64(Float64(J * -2.0) * sqrt(fma(Float64(Float64(0.25 * U_m) / J), Float64(U_m / J), 1.0)));
	else
		tmp = fma(Float64(J * J), Float64(2.0 / U_m), U_m);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -5e-246], N[(N[(J * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(J * J), $MachinePrecision] * N[(2.0 / U$95$m), $MachinePrecision] + U$95$m), $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 5.4%

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

   3. Taylor expanded in U around inf

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 51.7

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 51.7%

      \[\leadsto \color{blue}{-U} \]

   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))))) < -4.9999999999999997e-246

   1. Initial program 99.8%

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

   3. 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. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 57.9

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

   5. Applied rewrites 57.9%

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

   if -4.9999999999999997e-246 < (*.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 69.2%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left({J}^{2}, \frac{2}{U}, U\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto \mathsf{fma}\left(J \cdot J, \frac{2}{U}, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{-246}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot J, \frac{2}{U}, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot t\_0\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-246}:\\ \;\;\;\;J \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot J, \frac{2}{U\_m}, U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0))
          (* (* J -2.0) t_0))))
   (if (<= t_1 -5e+307)
     (- U_m)
     (if (<= t_1 -5e-246) (* J -2.0) (fma (* J J) (/ 2.0 U_m) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * ((J * -2.0) * t_0);
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = -U_m;
	} else if (t_1 <= -5e-246) {
		tmp = J * -2.0;
	} else {
		tmp = fma((J * J), (2.0 / U_m), U_m);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J * -2.0) * t_0))
	tmp = 0.0
	if (t_1 <= -5e+307)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-246)
		tmp = Float64(J * -2.0);
	else
		tmp = fma(Float64(J * J), Float64(2.0 / U_m), U_m);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+307], (-U$95$m), If[LessEqual[t$95$1, -5e-246], N[(J * -2.0), $MachinePrecision], N[(N[(J * J), $MachinePrecision] * N[(2.0 / U$95$m), $MachinePrecision] + U$95$m), $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))))) < -5e307

   1. Initial program 8.1%

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

   3. Taylor expanded in U around inf

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 53.0

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{-U} \]

   if -5e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999997e-246

   1. Initial program 99.8%

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

   3. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in K around 0

      \[\leadsto -2 \cdot \color{blue}{J} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.9%

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

   if -4.9999999999999997e-246 < (*.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 69.2%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in J around 0

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left({J}^{2}, \frac{2}{U}, U\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 27.2%

       \[\leadsto \mathsf{fma}\left(J \cdot J, \frac{2}{U}, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{-246}:\\ \;\;\;\;J \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot J, \frac{2}{U}, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot t\_0\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-246}:\\ \;\;\;\;J \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J) t_0)) 2.0) 1.0))
          (* (* J -2.0) t_0))))
   (if (<= t_1 -5e+307)
     (- U_m)
     (if (<= t_1 -5e-246) (* J -2.0) (* -1.0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * ((J * -2.0) * t_0);
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = -U_m;
	} else if (t_1 <= -5e-246) {
		tmp = J * -2.0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = sqrt((((u_m / ((2.0d0 * j) * t_0)) ** 2.0d0) + 1.0d0)) * ((j * (-2.0d0)) * t_0)
    if (t_1 <= (-5d+307)) then
        tmp = -u_m
    else if (t_1 <= (-5d-246)) then
        tmp = j * (-2.0d0)
    else
        tmp = (-1.0d0) * -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.sqrt((Math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * ((J * -2.0) * t_0);
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = -U_m;
	} else if (t_1 <= -5e-246) {
		tmp = J * -2.0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.sqrt((math.pow((U_m / ((2.0 * J) * t_0)), 2.0) + 1.0)) * ((J * -2.0) * t_0)
	tmp = 0
	if t_1 <= -5e+307:
		tmp = -U_m
	elif t_1 <= -5e-246:
		tmp = J * -2.0
	else:
		tmp = -1.0 * -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0) + 1.0)) * Float64(Float64(J * -2.0) * t_0))
	tmp = 0.0
	if (t_1 <= -5e+307)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-246)
		tmp = Float64(J * -2.0);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = sqrt((((U_m / ((2.0 * J) * t_0)) ^ 2.0) + 1.0)) * ((J * -2.0) * t_0);
	tmp = 0.0;
	if (t_1 <= -5e+307)
		tmp = -U_m;
	elseif (t_1 <= -5e-246)
		tmp = J * -2.0;
	else
		tmp = -1.0 * -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(J * -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+307], (-U$95$m), If[LessEqual[t$95$1, -5e-246], N[(J * -2.0), $MachinePrecision], N[(-1.0 * (-U$95$m)), $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))))) < -5e307

   1. Initial program 8.1%

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

   3. Taylor expanded in U around inf

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 53.0

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{-U} \]

   if -5e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999997e-246

   1. Initial program 99.8%

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

   3. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in K around 0

      \[\leadsto -2 \cdot \color{blue}{J} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.9%

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

   if -4.9999999999999997e-246 < (*.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 69.2%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in U around inf

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \left(-\color{blue}{U}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \leq -5 \cdot 10^{-246}:\\ \;\;\;\;J \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.1% accurate, 31.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 1.4 \cdot 10^{-31}:\\ \;\;\;\;J \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (if (<= U_m 1.4e-31) (* J -2.0) (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.4e-31) {
		tmp = J * -2.0;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (u_m <= 1.4d-31) then
        tmp = j * (-2.0d0)
    else
        tmp = -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 1.4e-31) {
		tmp = J * -2.0;
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if U_m <= 1.4e-31:
		tmp = J * -2.0
	else:
		tmp = -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 1.4e-31)
		tmp = Float64(J * -2.0);
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (U_m <= 1.4e-31)
		tmp = J * -2.0;
	else
		tmp = -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[U$95$m, 1.4e-31], N[(J * -2.0), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 1.3999999999999999e-31

   1. Initial program 80.7%

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

   3. Taylor expanded in U around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in K around 0

      \[\leadsto -2 \cdot \color{blue}{J} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.0%

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

   if 1.3999999999999999e-31 < U

   1. Initial program 48.0%

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

   3. Taylor expanded in U around inf

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 40.7

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 26.9% accurate, 124.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ -U\_m \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m) :precision binary64 (- U_m))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	return -U_m;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = -u_m
end function

Java

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

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	return -U_m

Julia

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

MATLAB

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

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := (-U$95$m)

Derivation

1. Initial program 71.4%

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

3. Taylor expanded in U around inf

   \[\leadsto \color{blue}{-1 \cdot U} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

   2. lower-neg.f64 24.8

      \[\leadsto \color{blue}{-U} \]

5. Applied rewrites 24.8%

   \[\leadsto \color{blue}{-U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024241 
(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)))))