Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.5% → 99.1%

Time: 14.0s

Alternatives: 17

Speedup: 0.7×

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: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 5e+300)
       t_1
       (*
        (- U_m)
        (fma
         -2.0
         (* (pow (cos (* K 0.5)) 2.0) (/ (* J J) (* U_m U_m)))
         -1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 5e+300) {
		tmp = t_1;
	} else {
		tmp = -U_m * fma(-2.0, (pow(cos((K * 0.5)), 2.0) * ((J * J) / (U_m * U_m))), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 5e+300)
		tmp = t_1;
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((cos(Float64(K * 0.5)) ^ 2.0) * Float64(Float64(J * J) / Float64(U_m * U_m))), -1.0));
	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[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 5e+300], t$95$1, N[((-U$95$m) * N[(-2.0 * N[(N[Power[N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < 5.00000000000000026e300

   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

   if 5.00000000000000026e300 < (*.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 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 39.5

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

   5. Applied rewrites 39.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+160}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-88}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot \frac{U\_m}{J}}{J}, 1\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (* (* -2.0 J) (cos (* K 0.5)))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -1e+160)
       t_2
       (if (<= t_1 -2e-88)
         (* (* -2.0 J) (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
         (if (<= t_1 -4e-178)
           (*
            (sqrt (fma 0.25 (/ (* U_m (/ U_m J)) J) 1.0))
            (*
             (* -2.0 J)
             (fma (* K K) (fma (* K K) 0.0026041666666666665 -0.125) 1.0)))
           t_2))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = (-2.0 * J) * cos((K * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e+160) {
		tmp = t_2;
	} else if (t_1 <= -2e-88) {
		tmp = (-2.0 * J) * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else if (t_1 <= -4e-178) {
		tmp = sqrt(fma(0.25, ((U_m * (U_m / J)) / J), 1.0)) * ((-2.0 * J) * fma((K * K), fma((K * K), 0.0026041666666666665, -0.125), 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+160)
		tmp = t_2;
	elseif (t_1 <= -2e-88)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J * J)), 1.0)));
	elseif (t_1 <= -4e-178)
		tmp = Float64(sqrt(fma(0.25, Float64(Float64(U_m * Float64(U_m / J)) / J), 1.0)) * Float64(Float64(-2.0 * J) * fma(Float64(K * K), fma(Float64(K * K), 0.0026041666666666665, -0.125), 1.0)));
	else
		tmp = t_2;
	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[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e+160], t$95$2, If[LessEqual[t$95$1, -2e-88], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-178], N[(N[Sqrt[N[(0.25 * N[(N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(-2.0 * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

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))))) < -inf.0

   1. Initial program 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < -1.00000000000000001e160 or -3.9999999999999998e-178 < (*.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 87.1%

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

   3. Taylor expanded in 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 63.3

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

   5. Applied rewrites 63.3%

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

   if -1.00000000000000001e160 < (*.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))))) < -1.99999999999999987e-88

   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 58.4

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

   5. Applied rewrites 58.4%

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

   if -1.99999999999999987e-88 < (*.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))))) < -3.9999999999999998e-178

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, {K}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      10. lower-*.f64 28.0

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   8. Applied rewrites 28.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   9. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{U}{J} \cdot \frac{U}{J}}, 1\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\frac{U}{J} \cdot U}}{J}, 1\right)} \]

      5. lower-/.f64 46.6

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{\color{blue}{\frac{U}{J}} \cdot U}{J}, 1\right)} \]

   10. Applied rewrites 46.6%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{+160}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-88}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -4 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot \frac{U}{J}}{J}, 1\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+233}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U\_m}{J}\right)}^{2}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + \frac{U\_m \cdot \frac{U\_m}{J \cdot 2}}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -2e+233)
       (* t_1 (sqrt (+ 1.0 (pow (* 0.5 (/ U_m J)) 2.0))))
       (if (<= t_2 5e+300)
         (*
          t_1
          (sqrt
           (+
            1.0
            (/
             (* U_m (/ U_m (* J 2.0)))
             (* (* J 2.0) (fma 0.5 (cos K) 0.5))))))
         (*
          (- U_m)
          (fma
           -2.0
           (* (pow (cos (* K 0.5)) 2.0) (/ (* J J) (* U_m U_m)))
           -1.0)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -2e+233) {
		tmp = t_1 * sqrt((1.0 + pow((0.5 * (U_m / J)), 2.0)));
	} else if (t_2 <= 5e+300) {
		tmp = t_1 * sqrt((1.0 + ((U_m * (U_m / (J * 2.0))) / ((J * 2.0) * fma(0.5, cos(K), 0.5)))));
	} else {
		tmp = -U_m * fma(-2.0, (pow(cos((K * 0.5)), 2.0) * ((J * J) / (U_m * U_m))), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -2e+233)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U_m / J)) ^ 2.0))));
	elseif (t_2 <= 5e+300)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + Float64(Float64(U_m * Float64(U_m / Float64(J * 2.0))) / Float64(Float64(J * 2.0) * fma(0.5, cos(K), 0.5))))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((cos(Float64(K * 0.5)) ^ 2.0) * Float64(Float64(J * J) / Float64(U_m * U_m))), -1.0));
	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[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -2e+233], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+300], N[(t$95$1 * N[Sqrt[N[(1.0 + N[(N[(U$95$m * N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(J * 2.0), $MachinePrecision] * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[Power[N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < -1.99999999999999995e233

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if -1.99999999999999995e233 < (*.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))))) < 5.00000000000000026e300

   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

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. associate-/r* N/A

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in K around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if 5.00000000000000026e300 < (*.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 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 39.5

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

   5. Applied rewrites 39.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+233}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \frac{U \cdot \frac{U}{J \cdot 2}}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-154}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U\_m}{J}\right)}^{2}}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot t\_0\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{\left(J \cdot J\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {t\_0}^{2} \cdot \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0))))))
   (if (<= t_3 (- INFINITY))
     (- U_m)
     (if (<= t_3 4e-154)
       (* t_2 (sqrt (+ 1.0 (pow (* 0.5 (/ U_m J)) 2.0))))
       (if (<= t_3 5e+300)
         (*
          -2.0
          (*
           (* J t_0)
           (sqrt
            (fma 0.25 (/ (* U_m U_m) (* (* J J) (fma 0.5 (cos K) 0.5))) 1.0))))
         (*
          (- U_m)
          (fma -2.0 (* (pow t_0 2.0) (/ (* J J) (* U_m U_m))) -1.0)))))))

C

U_m = fabs(U);
double code(double J, 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) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_3 <= 4e-154) {
		tmp = t_2 * sqrt((1.0 + pow((0.5 * (U_m / J)), 2.0)));
	} else if (t_3 <= 5e+300) {
		tmp = -2.0 * ((J * t_0) * sqrt(fma(0.25, ((U_m * U_m) / ((J * J) * fma(0.5, cos(K), 0.5))), 1.0)));
	} else {
		tmp = -U_m * fma(-2.0, (pow(t_0, 2.0) * ((J * J) / (U_m * U_m))), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_3 <= 4e-154)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U_m / J)) ^ 2.0))));
	elseif (t_3 <= 5e+300)
		tmp = Float64(-2.0 * Float64(Float64(J * t_0) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(Float64(J * J) * fma(0.5, cos(K), 0.5))), 1.0))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((t_0 ^ 2.0) * Float64(Float64(J * J) / Float64(U_m * U_m))), -1.0));
	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 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], (-U$95$m), If[LessEqual[t$95$3, 4e-154], N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+300], N[(-2.0 * N[(N[(J * t$95$0), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(N[(J * J), $MachinePrecision] * N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < 3.9999999999999999e-154

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if 3.9999999999999999e-154 < (*.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))))) < 5.00000000000000026e300

   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

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. associate-/r* N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in K around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 83.8%

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

   if 5.00000000000000026e300 < (*.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 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 39.5

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

   5. Applied rewrites 39.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 4 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{\left(J \cdot J\right) \cdot \mathsf{fma}\left(0.5, \cos K, 0.5\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-154}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U\_m}{J}\right)}^{2}}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(t\_0 \cdot \sqrt{\mathsf{fma}\left(U\_m, \frac{U\_m}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(\cos K, J, J\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {t\_0}^{2} \cdot \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0))))))
   (if (<= t_3 (- INFINITY))
     (- U_m)
     (if (<= t_3 4e-154)
       (* t_2 (sqrt (+ 1.0 (pow (* 0.5 (/ U_m J)) 2.0))))
       (if (<= t_3 5e+300)
         (*
          (* -2.0 J)
          (* t_0 (sqrt (fma U_m (/ U_m (* (* J 2.0) (fma (cos K) J J))) 1.0))))
         (*
          (- U_m)
          (fma -2.0 (* (pow t_0 2.0) (/ (* J J) (* U_m U_m))) -1.0)))))))

C

U_m = fabs(U);
double code(double J, 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) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_3 <= 4e-154) {
		tmp = t_2 * sqrt((1.0 + pow((0.5 * (U_m / J)), 2.0)));
	} else if (t_3 <= 5e+300) {
		tmp = (-2.0 * J) * (t_0 * sqrt(fma(U_m, (U_m / ((J * 2.0) * fma(cos(K), J, J))), 1.0)));
	} else {
		tmp = -U_m * fma(-2.0, (pow(t_0, 2.0) * ((J * J) / (U_m * U_m))), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_3 <= 4e-154)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U_m / J)) ^ 2.0))));
	elseif (t_3 <= 5e+300)
		tmp = Float64(Float64(-2.0 * J) * Float64(t_0 * sqrt(fma(U_m, Float64(U_m / Float64(Float64(J * 2.0) * fma(cos(K), J, J))), 1.0))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((t_0 ^ 2.0) * Float64(Float64(J * J) / Float64(U_m * U_m))), -1.0));
	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 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], (-U$95$m), If[LessEqual[t$95$3, 4e-154], N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+300], N[(N[(-2.0 * J), $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(U$95$m * N[(U$95$m / N[(N[(J * 2.0), $MachinePrecision] * N[(N[Cos[K], $MachinePrecision] * J + J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < 3.9999999999999999e-154

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if 3.9999999999999999e-154 < (*.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))))) < 5.00000000000000026e300

   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

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. associate-/r* N/A

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 89.2%

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

   if 5.00000000000000026e300 < (*.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 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 39.5

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

   5. Applied rewrites 39.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 4 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(\cos K, J, J\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-88}:\\ \;\;\;\;t\_2 \cdot \left(\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(U\_m \cdot 0.25, \frac{U\_m}{J \cdot J}, 1\right)}\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot \frac{U\_m}{J}}{J}, 1\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (cos (* K 0.5))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-88)
       (* t_2 (* (* -2.0 J) (sqrt (fma (* U_m 0.25) (/ U_m (* J J)) 1.0))))
       (if (<= t_1 -4e-178)
         (*
          (sqrt (fma 0.25 (/ (* U_m (/ U_m J)) J) 1.0))
          (*
           (* -2.0 J)
           (fma (* K K) (fma (* K K) 0.0026041666666666665 -0.125) 1.0)))
         (* (* -2.0 J) t_2))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = cos((K * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-88) {
		tmp = t_2 * ((-2.0 * J) * sqrt(fma((U_m * 0.25), (U_m / (J * J)), 1.0)));
	} else if (t_1 <= -4e-178) {
		tmp = sqrt(fma(0.25, ((U_m * (U_m / J)) / J), 1.0)) * ((-2.0 * J) * fma((K * K), fma((K * K), 0.0026041666666666665, -0.125), 1.0));
	} else {
		tmp = (-2.0 * J) * t_2;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-88)
		tmp = Float64(t_2 * Float64(Float64(-2.0 * J) * sqrt(fma(Float64(U_m * 0.25), Float64(U_m / Float64(J * J)), 1.0))));
	elseif (t_1 <= -4e-178)
		tmp = Float64(sqrt(fma(0.25, Float64(Float64(U_m * Float64(U_m / J)) / J), 1.0)) * Float64(Float64(-2.0 * J) * fma(Float64(K * K), fma(Float64(K * K), 0.0026041666666666665, -0.125), 1.0)));
	else
		tmp = Float64(Float64(-2.0 * J) * t_2);
	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[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-88], N[(t$95$2 * N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m * 0.25), $MachinePrecision] * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-178], N[(N[Sqrt[N[(0.25 * N[(N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(-2.0 * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J), $MachinePrecision] * t$95$2), $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))))) < -inf.0

   1. Initial program 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < -1.99999999999999987e-88

   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 74.1

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

   5. Applied rewrites 74.1%

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

   7. Applied rewrites 85.9%

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

   if -1.99999999999999987e-88 < (*.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))))) < -3.9999999999999998e-178

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, {K}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      10. lower-*.f64 28.0

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   8. Applied rewrites 28.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   9. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{U}{J} \cdot \frac{U}{J}}, 1\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\frac{U}{J} \cdot U}}{J}, 1\right)} \]

      5. lower-/.f64 46.6

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{\color{blue}{\frac{U}{J}} \cdot U}{J}, 1\right)} \]

   10. Applied rewrites 46.6%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

   if -3.9999999999999998e-178 < (*.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 82.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 56.9

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

   5. Applied rewrites 56.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-88}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(U \cdot 0.25, \frac{U}{J \cdot J}, 1\right)}\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -4 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot \frac{U}{J}}{J}, 1\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right) \cdot \left(\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(U\_m, 0.25 \cdot \frac{U\_m}{J \cdot J}, 1\right)}\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-156}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J \cdot J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(U\_m \cdot \left(K \cdot 0.125\right), K, -U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e+180)
       (*
        (fma K (* K (fma (* K K) 0.0026041666666666665 -0.125)) 1.0)
        (* (* -2.0 J) (sqrt (fma U_m (* 0.25 (/ U_m (* J J))) 1.0))))
       (if (<= t_1 -5e-156)
         (* (* -2.0 J) (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
         (fma (* U_m (* K 0.125)) K (- 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 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e+180) {
		tmp = fma(K, (K * fma((K * K), 0.0026041666666666665, -0.125)), 1.0) * ((-2.0 * J) * sqrt(fma(U_m, (0.25 * (U_m / (J * J))), 1.0)));
	} else if (t_1 <= -5e-156) {
		tmp = (-2.0 * J) * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else {
		tmp = fma((U_m * (K * 0.125)), K, -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(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e+180)
		tmp = Float64(fma(K, Float64(K * fma(Float64(K * K), 0.0026041666666666665, -0.125)), 1.0) * Float64(Float64(-2.0 * J) * sqrt(fma(U_m, Float64(0.25 * Float64(U_m / Float64(J * J))), 1.0))));
	elseif (t_1 <= -5e-156)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J * J)), 1.0)));
	else
		tmp = fma(Float64(U_m * Float64(K * 0.125)), K, 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[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e+180], N[(N[(K * N[(K * N[(N[(K * K), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(U$95$m * N[(0.25 * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-156], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(U$95$m * N[(K * 0.125), $MachinePrecision]), $MachinePrecision] * K + (-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))))) < -inf.0

   1. Initial program 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < -2e180

   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 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, {K}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      10. lower-*.f64 31.7

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   8. Applied rewrites 31.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   10. Applied rewrites 48.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right) \cdot \left(\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{J \cdot J} \cdot 0.25, 1\right)}\right)} \]

   if -2e180 < (*.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))))) < -5.00000000000000007e-156

   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 54.7

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

   5. Applied rewrites 54.7%

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

   if -5.00000000000000007e-156 < (*.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 83.1%

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

   3. Taylor expanded in 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 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in U around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 15.2

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-out N/A

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

       4. lower-*.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

          \[\leadsto U \cdot \mathsf{fma}\left(\frac{1}{8}, \color{blue}{K \cdot K}, -1\right) \]

       7. lower-*.f64 17.3

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

   11. Applied rewrites 17.3%

       \[\leadsto \color{blue}{U \cdot \mathsf{fma}\left(0.125, K \cdot K, -1\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. distribute-lft-in N/A

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

       3. *-commutative N/A

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

       4. neg-mul-1 N/A

          \[\leadsto U \cdot \left(\frac{1}{8} \cdot \left(K \cdot K\right)\right) + \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto U \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(K \cdot K\right)}\right) + \left(\mathsf{neg}\left(U\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto U \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot K\right) \cdot K\right)} + \left(\mathsf{neg}\left(U\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \color{blue}{\left(U \cdot \left(\frac{1}{8} \cdot K\right)\right) \cdot K} + \left(\mathsf{neg}\left(U\right)\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(U \cdot \left(\frac{1}{8} \cdot K\right), K, \mathsf{neg}\left(U\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{U \cdot \left(\frac{1}{8} \cdot K\right)}, K, \mathsf{neg}\left(U\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(U \cdot \color{blue}{\left(K \cdot \frac{1}{8}\right)}, K, \mathsf{neg}\left(U\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(U \cdot \color{blue}{\left(K \cdot \frac{1}{8}\right)}, K, \mathsf{neg}\left(U\right)\right) \]

       12. lower-neg.f64 17.4

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

   13. Applied rewrites 17.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(U \cdot \left(K \cdot 0.125\right), K, -U\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(K, K \cdot \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right) \cdot \left(\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(U, 0.25 \cdot \frac{U}{J \cdot J}, 1\right)}\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-156}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(U \cdot \left(K \cdot 0.125\right), K, -U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+180}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), J\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-156}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J \cdot J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(U\_m \cdot \left(K \cdot 0.125\right), K, -U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e+180)
       (*
        -2.0
        (fma (* J (* K K)) (fma (* K K) 0.0026041666666666665 -0.125) J))
       (if (<= t_1 -5e-156)
         (* (* -2.0 J) (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
         (fma (* U_m (* K 0.125)) K (- 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 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e+180) {
		tmp = -2.0 * fma((J * (K * K)), fma((K * K), 0.0026041666666666665, -0.125), J);
	} else if (t_1 <= -5e-156) {
		tmp = (-2.0 * J) * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else {
		tmp = fma((U_m * (K * 0.125)), K, -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(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e+180)
		tmp = Float64(-2.0 * fma(Float64(J * Float64(K * K)), fma(Float64(K * K), 0.0026041666666666665, -0.125), J));
	elseif (t_1 <= -5e-156)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J * J)), 1.0)));
	else
		tmp = fma(Float64(U_m * Float64(K * 0.125)), K, 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[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e+180], N[(-2.0 * N[(N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-156], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(U$95$m * N[(K * 0.125), $MachinePrecision]), $MachinePrecision] * K + (-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))))) < -inf.0

   1. Initial program 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < -2e180

   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 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, {K}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      10. lower-*.f64 31.7

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   8. Applied rewrites 31.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right) \]

       3. distribute-lft-in N/A

          \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) + J \cdot 1\right)} \]

       4. associate-*r* N/A

          \[\leadsto -2 \cdot \left(\color{blue}{\left(J \cdot {K}^{2}\right) \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)} + J \cdot 1\right) \]

       5. *-rgt-identity N/A

          \[\leadsto -2 \cdot \left(\left(J \cdot {K}^{2}\right) \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + \color{blue}{J}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(J \cdot {K}^{2}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, J\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto -2 \cdot \mathsf{fma}\left(\color{blue}{J \cdot {K}^{2}}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, J\right) \]

       8. unpow2 N/A

          \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \color{blue}{\left(K \cdot K\right)}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, J\right) \]

       9. lower-*.f64 N/A

          \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \color{blue}{\left(K \cdot K\right)}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, J\right) \]

       10. sub-neg N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \color{blue}{\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, J\right) \]

       11. *-commutative N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), J\right) \]

       12. metadata-eval N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), {K}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, J\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, J\right) \]

       14. unpow2 N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384}, \frac{-1}{8}\right), J\right) \]

       15. lower-*.f64 41.5

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \mathsf{fma}\left(\color{blue}{K \cdot K}, 0.0026041666666666665, -0.125\right), J\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \color{blue}{-2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), J\right)} \]

   if -2e180 < (*.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))))) < -5.00000000000000007e-156

   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 54.7

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

   5. Applied rewrites 54.7%

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

   if -5.00000000000000007e-156 < (*.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 83.1%

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

   3. Taylor expanded in 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 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in U around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 15.2

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in K around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-out N/A

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

       4. lower-*.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

          \[\leadsto U \cdot \mathsf{fma}\left(\frac{1}{8}, \color{blue}{K \cdot K}, -1\right) \]

       7. lower-*.f64 17.3

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

   11. Applied rewrites 17.3%

       \[\leadsto \color{blue}{U \cdot \mathsf{fma}\left(0.125, K \cdot K, -1\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. distribute-lft-in N/A

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

       3. *-commutative N/A

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

       4. neg-mul-1 N/A

          \[\leadsto U \cdot \left(\frac{1}{8} \cdot \left(K \cdot K\right)\right) + \color{blue}{\left(\mathsf{neg}\left(U\right)\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto U \cdot \left(\frac{1}{8} \cdot \color{blue}{\left(K \cdot K\right)}\right) + \left(\mathsf{neg}\left(U\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto U \cdot \color{blue}{\left(\left(\frac{1}{8} \cdot K\right) \cdot K\right)} + \left(\mathsf{neg}\left(U\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \color{blue}{\left(U \cdot \left(\frac{1}{8} \cdot K\right)\right) \cdot K} + \left(\mathsf{neg}\left(U\right)\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(U \cdot \left(\frac{1}{8} \cdot K\right), K, \mathsf{neg}\left(U\right)\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{U \cdot \left(\frac{1}{8} \cdot K\right)}, K, \mathsf{neg}\left(U\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(U \cdot \color{blue}{\left(K \cdot \frac{1}{8}\right)}, K, \mathsf{neg}\left(U\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(U \cdot \color{blue}{\left(K \cdot \frac{1}{8}\right)}, K, \mathsf{neg}\left(U\right)\right) \]

       12. lower-neg.f64 17.4

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

   13. Applied rewrites 17.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(U \cdot \left(K \cdot 0.125\right), K, -U\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+180}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), J\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-156}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(U \cdot \left(K \cdot 0.125\right), K, -U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot t\_2\right)\right) \cdot \sqrt{1 + {\left(\frac{t\_2 \cdot \left(J \cdot 2\right)}{U\_m}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {t\_2}^{2} \cdot \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (cos (* K 0.5))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 5e+300)
       (*
        (* J (* -2.0 t_2))
        (sqrt (+ 1.0 (pow (/ (* t_2 (* J 2.0)) U_m) -2.0))))
       (*
        (- U_m)
        (fma -2.0 (* (pow t_2 2.0) (/ (* J J) (* U_m U_m))) -1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = cos((K * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 5e+300) {
		tmp = (J * (-2.0 * t_2)) * sqrt((1.0 + pow(((t_2 * (J * 2.0)) / U_m), -2.0)));
	} else {
		tmp = -U_m * fma(-2.0, (pow(t_2, 2.0) * ((J * J) / (U_m * U_m))), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 5e+300)
		tmp = Float64(Float64(J * Float64(-2.0 * t_2)) * sqrt(Float64(1.0 + (Float64(Float64(t_2 * Float64(J * 2.0)) / U_m) ^ -2.0))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((t_2 ^ 2.0) * Float64(Float64(J * J) / Float64(U_m * U_m))), -1.0));
	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[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 5e+300], N[(N[(J * N[(-2.0 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(N[(t$95$2 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < 5.00000000000000026e300

   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

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-/r* N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 95.7%

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if 5.00000000000000026e300 < (*.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 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 39.5

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

   5. Applied rewrites 39.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{1 + {\left(\frac{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)}{U}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U\_m}{J}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 5e+300)
       (* t_1 (sqrt (+ 1.0 (pow (* 0.5 (/ U_m J)) 2.0))))
       (*
        (- U_m)
        (fma
         -2.0
         (* (pow (cos (* K 0.5)) 2.0) (/ (* J J) (* U_m U_m)))
         -1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+300) {
		tmp = t_1 * sqrt((1.0 + pow((0.5 * (U_m / J)), 2.0)));
	} else {
		tmp = -U_m * fma(-2.0, (pow(cos((K * 0.5)), 2.0) * ((J * J) / (U_m * U_m))), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+300)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(0.5 * Float64(U_m / J)) ^ 2.0))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((cos(Float64(K * 0.5)) ^ 2.0) * Float64(Float64(J * J) / Float64(U_m * U_m))), -1.0));
	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[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+300], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(0.5 * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[Power[N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < 5.00000000000000026e300

   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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if 5.00000000000000026e300 < (*.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 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 39.5

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

   5. Applied rewrites 39.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot \frac{U\_m}{J}}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 5e+300)
       (* t_1 (sqrt (fma 0.25 (/ (* U_m (/ U_m J)) J) 1.0)))
       (*
        (- U_m)
        (fma
         -2.0
         (* (pow (cos (* K 0.5)) 2.0) (/ (* J J) (* U_m U_m)))
         -1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+300) {
		tmp = t_1 * sqrt(fma(0.25, ((U_m * (U_m / J)) / J), 1.0));
	} else {
		tmp = -U_m * fma(-2.0, (pow(cos((K * 0.5)), 2.0) * ((J * J) / (U_m * U_m))), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+300)
		tmp = Float64(t_1 * sqrt(fma(0.25, Float64(Float64(U_m * Float64(U_m / J)) / J), 1.0)));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((cos(Float64(K * 0.5)) ^ 2.0) * Float64(Float64(J * J) / Float64(U_m * U_m))), -1.0));
	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[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+300], N[(t$95$1 * N[Sqrt[N[(0.25 * N[(N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[Power[N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < 5.00000000000000026e300

   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 67.2

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

   5. Applied rewrites 67.2%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 85.2

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

   7. Applied rewrites 85.2%

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

   if 5.00000000000000026e300 < (*.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 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. 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 39.5

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

   5. Applied rewrites 39.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot \frac{U}{J}}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot \frac{U\_m}{J}}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m \cdot \frac{-\cos \left(K \cdot 0.5\right)}{\sqrt{\mathsf{fma}\left(0.5, \cos K, 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 5e+300)
       (* t_1 (sqrt (fma 0.25 (/ (* U_m (/ U_m J)) J) 1.0)))
       (* U_m (/ (- (cos (* K 0.5))) (sqrt (fma 0.5 (cos K) 0.5))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+300) {
		tmp = t_1 * sqrt(fma(0.25, ((U_m * (U_m / J)) / J), 1.0));
	} else {
		tmp = U_m * (-cos((K * 0.5)) / sqrt(fma(0.5, cos(K), 0.5)));
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+300)
		tmp = Float64(t_1 * sqrt(fma(0.25, Float64(Float64(U_m * Float64(U_m / J)) / J), 1.0)));
	else
		tmp = Float64(U_m * Float64(Float64(-cos(Float64(K * 0.5))) / sqrt(fma(0.5, cos(K), 0.5))));
	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[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+300], N[(t$95$1 * N[Sqrt[N[(0.25 * N[(N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(U$95$m * N[((-N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]) / N[Sqrt[N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision]], $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 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < 5.00000000000000026e300

   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 67.2

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

   5. Applied rewrites 67.2%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 85.2

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

   7. Applied rewrites 85.2%

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

   if 5.00000000000000026e300 < (*.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 5.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. associate-/r* N/A

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

   4. Applied rewrites 5.5%

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

   5. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-cos.f64 61.0

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

   7. Applied rewrites 61.0%

      \[\leadsto \color{blue}{-\left(U \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, \cos K, 0.5\right)}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-/.f64 N/A

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

   9. Applied rewrites 61.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot \frac{U}{J}}{J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U \cdot \frac{-\cos \left(K \cdot 0.5\right)}{\sqrt{\mathsf{fma}\left(0.5, \cos K, 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot \frac{U\_m}{J}}{J}, 1\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -4e-213)
       (*
        (sqrt (fma 0.25 (/ (* U_m (/ U_m J)) J) 1.0))
        (*
         (* -2.0 J)
         (fma (* K K) (fma (* K K) 0.0026041666666666665 -0.125) 1.0)))
       (* (- U_m) (cos (* K 0.5)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -4e-213) {
		tmp = sqrt(fma(0.25, ((U_m * (U_m / J)) / J), 1.0)) * ((-2.0 * J) * fma((K * K), fma((K * K), 0.0026041666666666665, -0.125), 1.0));
	} else {
		tmp = -U_m * cos((K * 0.5));
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -4e-213)
		tmp = Float64(sqrt(fma(0.25, Float64(Float64(U_m * Float64(U_m / J)) / J), 1.0)) * Float64(Float64(-2.0 * J) * fma(Float64(K * K), fma(Float64(K * K), 0.0026041666666666665, -0.125), 1.0)));
	else
		tmp = Float64(Float64(-U_m) * cos(Float64(K * 0.5)));
	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[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -4e-213], N[(N[Sqrt[N[(0.25 * N[(N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(-2.0 * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[Cos[N[(K * 0.5), $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 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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))))) < -3.9999999999999998e-213

   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 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, {K}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      10. lower-*.f64 36.1

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   8. Applied rewrites 36.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   9. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{U}{J} \cdot \frac{U}{J}}, 1\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\frac{U}{J} \cdot U}}{J}, 1\right)} \]

      5. lower-/.f64 44.2

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{\color{blue}{\frac{U}{J}} \cdot U}{J}, 1\right)} \]

   10. Applied rewrites 44.2%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

   if -3.9999999999999998e-213 < (*.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 82.0%

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

   3. Taylor expanded in 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 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 20.0

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

   8. Applied rewrites 20.0%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -4 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot \frac{U}{J}}{J}, 1\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	tmp = 0.0
	if (Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(-U_m);
	else
		tmp = Float64(t_1 * sqrt(fma(0.25, Float64(Float64(U_m * Float64(U_m / J)) / J), 1.0)));
	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[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], (-U$95$m), N[(t$95$1 * N[Sqrt[N[(0.25 * N[(N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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. Initial program 90.2%

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

   3. Taylor expanded in 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 60.9

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

   5. Applied rewrites 60.9%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 77.0

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

   7. Applied rewrites 77.0%

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

4. Final simplification 74.8%

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

Alternative 15: 50.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot \frac{U\_m}{J}}{J}, 1\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<=
        (*
         (* (* -2.0 J) t_0)
         (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))
        (- INFINITY))
     (- U_m)
     (*
      (sqrt (fma 0.25 (/ (* U_m (/ U_m J)) J) 1.0))
      (*
       (* -2.0 J)
       (fma (* K K) (fma (* K K) 0.0026041666666666665 -0.125) 1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)))) <= -((double) INFINITY)) {
		tmp = -U_m;
	} else {
		tmp = sqrt(fma(0.25, ((U_m * (U_m / J)) / J), 1.0)) * ((-2.0 * J) * fma((K * K), fma((K * K), 0.0026041666666666665, -0.125), 1.0));
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(-U_m);
	else
		tmp = Float64(sqrt(fma(0.25, Float64(Float64(U_m * Float64(U_m / J)) / J), 1.0)) * Float64(Float64(-2.0 * J) * fma(Float64(K * K), fma(Float64(K * K), 0.0026041666666666665, -0.125), 1.0)));
	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]}, If[LessEqual[N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], (-U$95$m), N[(N[Sqrt[N[(0.25 * N[(N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] / J), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(-2.0 * J), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   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. Initial program 90.2%

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

   3. Taylor expanded in 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 60.9

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, {K}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      10. lower-*.f64 32.4

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   8. Applied rewrites 32.4%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   9. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{U}{J} \cdot \frac{U}{J}}, 1\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{\color{blue}{\frac{U}{J} \cdot U}}{J}, 1\right)} \]

      5. lower-/.f64 43.6

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{\color{blue}{\frac{U}{J}} \cdot U}{J}, 1\right)} \]

   10. Applied rewrites 43.6%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{\frac{U}{J} \cdot U}{J}}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot \frac{U}{J}}{J}, 1\right)} \cdot \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;U\_m \leq 5.2 \cdot 10^{+17}:\\ \;\;\;\;-2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), J\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= U_m 5.2e+17)
   (* -2.0 (fma (* J (* K K)) (fma (* K K) 0.0026041666666666665 -0.125) J))
   (- U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (U_m <= 5.2e+17) {
		tmp = -2.0 * fma((J * (K * K)), fma((K * K), 0.0026041666666666665, -0.125), J);
	} else {
		tmp = -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (U_m <= 5.2e+17)
		tmp = Float64(-2.0 * fma(Float64(J * Float64(K * K)), fma(Float64(K * K), 0.0026041666666666665, -0.125), J));
	else
		tmp = Float64(-U_m);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[U$95$m, 5.2e+17], N[(-2.0 * N[(N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision], (-U$95$m)]

Derivation

1. Split input into 2 regimes

2. if U < 5.2e17

   1. Initial program 87.0%

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

   3. Taylor expanded in 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 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      5. sub-neg N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, {K}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384}, \frac{-1}{8}\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

      10. lower-*.f64 32.7

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(\color{blue}{K \cdot K}, 0.0026041666666666665, -0.125\right), 1\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   8. Applied rewrites 32.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)} \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \left(1 + {K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + 1\right)}\right) \]

       3. distribute-lft-in N/A

          \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \left({K}^{2} \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)\right) + J \cdot 1\right)} \]

       4. associate-*r* N/A

          \[\leadsto -2 \cdot \left(\color{blue}{\left(J \cdot {K}^{2}\right) \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right)} + J \cdot 1\right) \]

       5. *-rgt-identity N/A

          \[\leadsto -2 \cdot \left(\left(J \cdot {K}^{2}\right) \cdot \left(\frac{1}{384} \cdot {K}^{2} - \frac{1}{8}\right) + \color{blue}{J}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\mathsf{fma}\left(J \cdot {K}^{2}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, J\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto -2 \cdot \mathsf{fma}\left(\color{blue}{J \cdot {K}^{2}}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, J\right) \]

       8. unpow2 N/A

          \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \color{blue}{\left(K \cdot K\right)}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, J\right) \]

       9. lower-*.f64 N/A

          \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \color{blue}{\left(K \cdot K\right)}, \frac{1}{384} \cdot {K}^{2} - \frac{1}{8}, J\right) \]

       10. sub-neg N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \color{blue}{\frac{1}{384} \cdot {K}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, J\right) \]

       11. *-commutative N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \color{blue}{{K}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), J\right) \]

       12. metadata-eval N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), {K}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, J\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, J\right) \]

       14. unpow2 N/A

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{1}{384}, \frac{-1}{8}\right), J\right) \]

       15. lower-*.f64 31.9

           \[\leadsto -2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \mathsf{fma}\left(\color{blue}{K \cdot K}, 0.0026041666666666665, -0.125\right), J\right) \]

   11. Applied rewrites 31.9%

       \[\leadsto \color{blue}{-2 \cdot \mathsf{fma}\left(J \cdot \left(K \cdot K\right), \mathsf{fma}\left(K \cdot K, 0.0026041666666666665, -0.125\right), J\right)} \]

   if 5.2e17 < U

   1. Initial program 51.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 49.8

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

   5. Applied rewrites 49.8%

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

Alternative 17: 26.6% 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 80.0%

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

3. Taylor expanded in 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 26.8

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

5. Applied rewrites 26.8%

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

Reproduce

herbie shell --seed 2024214 
(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)))))