Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.1% → 99.6%

Time: 14.1s

Alternatives: 19

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: 73.1% 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

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot \left(J\_m \cdot t\_0\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(J\_m \cdot 2\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {t\_0}^{2} \cdot \frac{J\_m \cdot J\_m}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 1e+304)
        (*
         (* -2.0 (* J_m t_0))
         (sqrt (+ 1.0 (pow (/ U_m (* (* J_m 2.0) t_0)) 2.0))))
        (*
         (- U_m)
         (fma -2.0 (* (pow t_0 2.0) (/ (* J_m J_m) (* U_m U_m))) -1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / (t_1 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+304) {
		tmp = (-2.0 * (J_m * t_0)) * sqrt((1.0 + pow((U_m / ((J_m * 2.0) * t_0)), 2.0)));
	} else {
		tmp = -U_m * fma(-2.0, (pow(t_0, 2.0) * ((J_m * J_m) / (U_m * U_m))), -1.0);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+304)
		tmp = Float64(Float64(-2.0 * Float64(J_m * t_0)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(J_m * 2.0) * t_0)) ^ 2.0))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((t_0 ^ 2.0) * Float64(Float64(J_m * J_m) / Float64(U_m * U_m))), -1.0));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+304], N[(N[(-2.0 * N[(J$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(J$95$m * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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))))) < 9.9999999999999994e303

   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. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 9.9999999999999994e303 < (*.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 5.8%

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

   3. Taylor expanded in U around -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. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+115}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(U\_m, U\_m \cdot \frac{0.125}{J\_m \cdot J\_m}, 1\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-50}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-233}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \frac{\mathsf{fma}\left(U\_m, 0.5, \frac{J\_m \cdot J\_m}{U\_m}\right)}{J\_m}\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \frac{U\_m \cdot -0.5}{J\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 -5e+115)
        (* t_1 (fma U_m (* U_m (/ 0.125 (* J_m J_m))) 1.0))
        (if (<= t_2 -2e-50)
          (* (* -2.0 J_m) (sqrt (fma 0.25 (/ (* U_m U_m) (* J_m J_m)) 1.0)))
          (if (<= t_2 -2e-233)
            (* (* -2.0 J_m) (/ (fma U_m 0.5 (/ (* J_m J_m) U_m)) J_m))
            (if (<= t_2 1e+304)
              (* (* -2.0 J_m) (cos (* K 0.5)))
              (* (* -2.0 J_m) (/ (* U_m -0.5) J_m))))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -5e+115) {
		tmp = t_1 * fma(U_m, (U_m * (0.125 / (J_m * J_m))), 1.0);
	} else if (t_2 <= -2e-50) {
		tmp = (-2.0 * J_m) * sqrt(fma(0.25, ((U_m * U_m) / (J_m * J_m)), 1.0));
	} else if (t_2 <= -2e-233) {
		tmp = (-2.0 * J_m) * (fma(U_m, 0.5, ((J_m * J_m) / U_m)) / J_m);
	} else if (t_2 <= 1e+304) {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	} else {
		tmp = (-2.0 * J_m) * ((U_m * -0.5) / J_m);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -5e+115)
		tmp = Float64(t_1 * fma(U_m, Float64(U_m * Float64(0.125 / Float64(J_m * J_m))), 1.0));
	elseif (t_2 <= -2e-50)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 1.0)));
	elseif (t_2 <= -2e-233)
		tmp = Float64(Float64(-2.0 * J_m) * Float64(fma(U_m, 0.5, Float64(Float64(J_m * J_m) / U_m)) / J_m));
	elseif (t_2 <= 1e+304)
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	else
		tmp = Float64(Float64(-2.0 * J_m) * Float64(Float64(U_m * -0.5) / J_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -5e+115], N[(t$95$1 * N[(U$95$m * N[(U$95$m * N[(0.125 / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-50], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-233], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(U$95$m * 0.5 + N[(N[(J$95$m * J$95$m), $MachinePrecision] / U$95$m), $MachinePrecision]), $MachinePrecision] / J$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+304], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[(N[(U$95$m * -0.5), $MachinePrecision] / J$95$m), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 6 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 5.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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))))) < -5.00000000000000008e115

   1. Initial program 99.9%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 79.5%

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

   if -5.00000000000000008e115 < (*.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))))) < -2.00000000000000002e-50

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if -2.00000000000000002e-50 < (*.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))))) < -1.99999999999999992e-233

   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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 40.8

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

   8. Applied rewrites 40.8%

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

   9. Taylor expanded in J around 0

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

   11. Applied rewrites 50.8%

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

   if -1.99999999999999992e-233 < (*.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))))) < 9.9999999999999994e303

   1. Initial program 99.7%

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

   3. Taylor expanded in J around inf

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} \]
   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

   if 9.9999999999999994e303 < (*.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 8.7%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 7.4

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in K around 0

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

      1. lower-*.f64 0.3

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

   8. Applied rewrites 0.3%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 57.9%

       \[\leadsto \left(-2 \cdot J\right) \cdot \frac{U \cdot -0.5}{\color{blue}{J}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+115}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(U, U \cdot \frac{0.125}{J \cdot J}, 1\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-50}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-233}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{\mathsf{fma}\left(U, 0.5, \frac{J \cdot J}{U}\right)}{J}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+304}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \frac{U \cdot -0.5}{J}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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)))))