Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 72.3% → 99.0%

Time: 12.6s

Alternatives: 16

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: 72.3% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+286}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J\_m \cdot J\_m}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 2e+286)
        t_1
        (*
         (- U_m)
         (fma
          -2.0
          (* (pow (cos (* K 0.5)) 2.0) (/ (* J_m J_m) (* U_m U_m)))
          -1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e+286) {
		tmp = t_1;
	} else {
		tmp = -U_m * fma(-2.0, (pow(cos((K * 0.5)), 2.0) * ((J_m * J_m) / (U_m * U_m))), -1.0);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e+286)
		tmp = t_1;
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((cos(Float64(K * 0.5)) ^ 2.0) * Float64(Float64(J_m * J_m) / Float64(U_m * U_m))), -1.0));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 2e+286], 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$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   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))))) < 2.00000000000000007e286

   1. Initial program 99.9%

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

   if 2.00000000000000007e286 < (*.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 11.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 48.7

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

   5. Applied rewrites 48.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\_m\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-63}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(J\_m \cdot 2\right) \cdot 1}\right)}^{2}}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot t\_0\right) \cdot \sqrt{\mathsf{fma}\left(U\_m, \frac{U\_m}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(\left(J\_m \cdot J\_m\right) \cdot 4\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-U\_m\right) \cdot t\_0}{\sqrt{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J_m) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_3 -2e+295)
      (- U_m)
      (if (<= t_3 2e-63)
        (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* (* J_m 2.0) 1.0)) 2.0))))
        (if (<= t_3 5e+265)
          (*
           J_m
           (*
            (* -2.0 t_0)
            (sqrt
             (fma
              U_m
              (/ U_m (* (fma 0.5 (cos K) 0.5) (* (* J_m J_m) 4.0)))
              1.0))))
          (/ (* (- U_m) t_0) (sqrt (fma (cos K) 0.5 0.5)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J_m) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / (t_1 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_3 <= -2e+295) {
		tmp = -U_m;
	} else if (t_3 <= 2e-63) {
		tmp = t_2 * sqrt((1.0 + pow((U_m / ((J_m * 2.0) * 1.0)), 2.0)));
	} else if (t_3 <= 5e+265) {
		tmp = J_m * ((-2.0 * t_0) * sqrt(fma(U_m, (U_m / (fma(0.5, cos(K), 0.5) * ((J_m * J_m) * 4.0))), 1.0)));
	} else {
		tmp = (-U_m * t_0) / sqrt(fma(cos(K), 0.5, 0.5));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J_m) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -2e+295)
		tmp = Float64(-U_m);
	elseif (t_3 <= 2e-63)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(J_m * 2.0) * 1.0)) ^ 2.0))));
	elseif (t_3 <= 5e+265)
		tmp = Float64(J_m * Float64(Float64(-2.0 * t_0) * sqrt(fma(U_m, Float64(U_m / Float64(fma(0.5, cos(K), 0.5) * Float64(Float64(J_m * J_m) * 4.0))), 1.0))));
	else
		tmp = Float64(Float64(Float64(-U_m) * t_0) / sqrt(fma(cos(K), 0.5, 0.5)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J$95$m), $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$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, -2e+295], (-U$95$m), If[LessEqual[t$95$3, 2e-63], N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(J$95$m * 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+265], N[(J$95$m * N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[Sqrt[N[(U$95$m * N[(U$95$m / N[(N[(0.5 * N[Cos[K], $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(J$95$m * J$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-U$95$m) * t$95$0), $MachinePrecision] / N[Sqrt[N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $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))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.00000000000000013e-63

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 84.4%

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

   if 2.00000000000000013e-63 < (*.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))))) < 5.0000000000000002e265

   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

   4. Applied rewrites 94.5%

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

   5. Taylor expanded in K around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 94.5

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(\color{blue}{\left(J \cdot J\right)} \cdot 4\right)}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot J \]

   7. Applied rewrites 94.5%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(\left(J \cdot J\right) \cdot 4\right)}}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot J \]

   if 5.0000000000000002e265 < (*.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 18.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 13.7%

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

   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 50.2

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

   7. Applied rewrites 50.2%

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

   9. Applied rewrites 50.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J \cdot 2\right) \cdot 1}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{\mathsf{fma}\left(0.5, \cos K, 0.5\right) \cdot \left(\left(J \cdot J\right) \cdot 4\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-U\right) \cdot \cos \left(K \cdot 0.5\right)}{\sqrt{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot J\_m\right) \cdot t\_1\\ t_3 := t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-63}:\\ \;\;\;\;t\_2 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(J\_m \cdot 2\right) \cdot 1}\right)}^{2}}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+231}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot t\_0\right) \cdot \sqrt{\mathsf{fma}\left(0.5, \frac{U\_m \cdot U\_m}{J\_m \cdot \mathsf{fma}\left(J\_m, \cos K, J\_m\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-U\_m\right) \cdot t\_0}{\sqrt{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 J_m) t_1))
        (t_3 (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_3 -2e+295)
      (- U_m)
      (if (<= t_3 2e-63)
        (* t_2 (sqrt (+ 1.0 (pow (/ U_m (* (* J_m 2.0) 1.0)) 2.0))))
        (if (<= t_3 4e+231)
          (*
           J_m
           (*
            (* -2.0 t_0)
            (sqrt
             (fma 0.5 (/ (* U_m U_m) (* J_m (fma J_m (cos K) J_m))) 1.0))))
          (/ (* (- U_m) t_0) (sqrt (fma (cos K) 0.5 0.5)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * J_m) * t_1;
	double t_3 = t_2 * sqrt((1.0 + pow((U_m / (t_1 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_3 <= -2e+295) {
		tmp = -U_m;
	} else if (t_3 <= 2e-63) {
		tmp = t_2 * sqrt((1.0 + pow((U_m / ((J_m * 2.0) * 1.0)), 2.0)));
	} else if (t_3 <= 4e+231) {
		tmp = J_m * ((-2.0 * t_0) * sqrt(fma(0.5, ((U_m * U_m) / (J_m * fma(J_m, cos(K), J_m))), 1.0)));
	} else {
		tmp = (-U_m * t_0) / sqrt(fma(cos(K), 0.5, 0.5));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * J_m) * t_1)
	t_3 = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -2e+295)
		tmp = Float64(-U_m);
	elseif (t_3 <= 2e-63)
		tmp = Float64(t_2 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(J_m * 2.0) * 1.0)) ^ 2.0))));
	elseif (t_3 <= 4e+231)
		tmp = Float64(J_m * Float64(Float64(-2.0 * t_0) * sqrt(fma(0.5, Float64(Float64(U_m * U_m) / Float64(J_m * fma(J_m, cos(K), J_m))), 1.0))));
	else
		tmp = Float64(Float64(Float64(-U_m) * t_0) / sqrt(fma(cos(K), 0.5, 0.5)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J$95$m), $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$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, -2e+295], (-U$95$m), If[LessEqual[t$95$3, 2e-63], N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(J$95$m * 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+231], N[(J$95$m * N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[Sqrt[N[(0.5 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * N[(J$95$m * N[Cos[K], $MachinePrecision] + J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-U$95$m) * t$95$0), $MachinePrecision] / N[Sqrt[N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $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))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.00000000000000013e-63

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 84.4%

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

   if 2.00000000000000013e-63 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.0000000000000002e231

   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

   4. Applied rewrites 98.1%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 91.8

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

   9. Applied rewrites 91.8%

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

   if 4.0000000000000002e231 < (*.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 37.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 31.8%

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

   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 43.2

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

   7. Applied rewrites 43.2%

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

   9. Applied rewrites 43.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J \cdot 2\right) \cdot 1}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 4 \cdot 10^{+231}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.5, \frac{U \cdot U}{J \cdot \mathsf{fma}\left(J, \cos K, J\right)}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-U\right) \cdot \cos \left(K \cdot 0.5\right)}{\sqrt{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{+59}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(U\_m, U\_m \cdot \frac{0.125}{J\_m \cdot J\_m}, 1\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U\_m \cdot U\_m}{J\_m}}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 -2e+295)
      (- U_m)
      (if (<= t_2 -5e+59)
        (* t_1 (fma U_m (* U_m (/ 0.125 (* J_m J_m))) 1.0))
        (if (<= t_2 -2e-168)
          (* (sqrt (fma 0.25 (/ (/ (* U_m U_m) J_m) J_m) 1.0)) (* -2.0 J_m))
          (* (* -2.0 J_m) (cos (* K 0.5)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -2e+295) {
		tmp = -U_m;
	} else if (t_2 <= -5e+59) {
		tmp = t_1 * fma(U_m, (U_m * (0.125 / (J_m * J_m))), 1.0);
	} else if (t_2 <= -2e-168) {
		tmp = sqrt(fma(0.25, (((U_m * U_m) / J_m) / J_m), 1.0)) * (-2.0 * J_m);
	} else {
		tmp = (-2.0 * J_m) * cos((K * 0.5));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -2e+295)
		tmp = Float64(-U_m);
	elseif (t_2 <= -5e+59)
		tmp = Float64(t_1 * fma(U_m, Float64(U_m * Float64(0.125 / Float64(J_m * J_m))), 1.0));
	elseif (t_2 <= -2e-168)
		tmp = Float64(sqrt(fma(0.25, Float64(Float64(Float64(U_m * U_m) / J_m) / J_m), 1.0)) * Float64(-2.0 * J_m));
	else
		tmp = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+295], (-U$95$m), If[LessEqual[t$95$2, -5e+59], N[(t$95$1 * N[(U$95$m * N[(U$95$m * N[(0.125 / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-168], N[(N[Sqrt[N[(0.25 * N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $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))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999997e59

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in U around 0

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

   8. Applied rewrites 76.5%

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

   if -4.9999999999999997e59 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e-168

   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 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 78.3%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.9

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

   10. Applied rewrites 60.9%

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

   if -2.0000000000000001e-168 < (*.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 78.1%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 51.3

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

   5. Applied rewrites 51.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+59}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{fma}\left(U, U \cdot \frac{0.125}{J \cdot J}, 1\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U \cdot U}{J}}{J}, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ t_2 := \left(-2 \cdot J\_m\right) \cdot \cos \left(K \cdot 0.5\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+59}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U\_m \cdot U\_m}{J\_m}}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0)))))
        (t_2 (* (* -2.0 J_m) (cos (* K 0.5)))))
   (*
    J_s
    (if (<= t_1 -2e+295)
      (- U_m)
      (if (<= t_1 -5e+59)
        t_2
        (if (<= t_1 -2e-168)
          (* (sqrt (fma 0.25 (/ (/ (* U_m U_m) J_m) J_m) 1.0)) (* -2.0 J_m))
          t_2))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double t_2 = (-2.0 * J_m) * cos((K * 0.5));
	double tmp;
	if (t_1 <= -2e+295) {
		tmp = -U_m;
	} else if (t_1 <= -5e+59) {
		tmp = t_2;
	} else if (t_1 <= -2e-168) {
		tmp = sqrt(fma(0.25, (((U_m * U_m) / J_m) / J_m), 1.0)) * (-2.0 * J_m);
	} else {
		tmp = t_2;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	t_2 = Float64(Float64(-2.0 * J_m) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (t_1 <= -2e+295)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e+59)
		tmp = t_2;
	elseif (t_1 <= -2e-168)
		tmp = Float64(sqrt(fma(0.25, Float64(Float64(Float64(U_m * U_m) / J_m) / J_m), 1.0)) * Float64(-2.0 * J_m));
	else
		tmp = t_2;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+295], (-U$95$m), If[LessEqual[t$95$1, -5e+59], t$95$2, If[LessEqual[t$95$1, -2e-168], N[(N[Sqrt[N[(0.25 * N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $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))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.9999999999999997e59 or -2.0000000000000001e-168 < (*.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 83.8%

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

   3. Taylor expanded in J around 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 57.9

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

   5. Applied rewrites 57.9%

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

   if -4.9999999999999997e59 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e-168

   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 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 78.3%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.9

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

   10. Applied rewrites 60.9%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+59}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U \cdot U}{J}}{J}, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+286}:\\ \;\;\;\;J\_m \cdot \left(\sqrt{\mathsf{fma}\left(\frac{U\_m}{\mathsf{fma}\left(\cos K, J\_m, J\_m\right)}, \frac{U\_m}{J\_m \cdot 2}, 1\right)} \cdot \left(-2 \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, {t\_0}^{2} \cdot \frac{J\_m \cdot J\_m}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 -2e+295)
      (- U_m)
      (if (<= t_2 2e+286)
        (*
         J_m
         (*
          (sqrt (fma (/ U_m (fma (cos K) J_m J_m)) (/ U_m (* J_m 2.0)) 1.0))
          (* -2.0 t_0)))
        (*
         (- U_m)
         (fma -2.0 (* (pow t_0 2.0) (/ (* J_m J_m) (* U_m U_m))) -1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / (t_1 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -2e+295) {
		tmp = -U_m;
	} else if (t_2 <= 2e+286) {
		tmp = J_m * (sqrt(fma((U_m / fma(cos(K), J_m, J_m)), (U_m / (J_m * 2.0)), 1.0)) * (-2.0 * t_0));
	} else {
		tmp = -U_m * fma(-2.0, (pow(t_0, 2.0) * ((J_m * J_m) / (U_m * U_m))), -1.0);
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -2e+295)
		tmp = Float64(-U_m);
	elseif (t_2 <= 2e+286)
		tmp = Float64(J_m * Float64(sqrt(fma(Float64(U_m / fma(cos(K), J_m, J_m)), Float64(U_m / Float64(J_m * 2.0)), 1.0)) * Float64(-2.0 * t_0)));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64((t_0 ^ 2.0) * Float64(Float64(J_m * J_m) / Float64(U_m * U_m))), -1.0));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+295], (-U$95$m), If[LessEqual[t$95$2, 2e+286], N[(J$95$m * N[(N[Sqrt[N[(N[(U$95$m / N[(N[Cos[K], $MachinePrecision] * J$95$m + J$95$m), $MachinePrecision]), $MachinePrecision] * N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.00000000000000007e286

   1. Initial program 99.9%

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

   4. Applied rewrites 80.6%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 99.7

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity 99.7

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

      5. lift-*.f64 N/A

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

      6. *-rgt-identity 99.7

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

   8. Applied rewrites 99.7%

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

   if 2.00000000000000007e286 < (*.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 11.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 48.7

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

   5. Applied rewrites 48.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;J \cdot \left(\sqrt{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(\cos K, J, J\right)}, \frac{U}{J \cdot 2}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, {\cos \left(K \cdot 0.5\right)}^{2} \cdot \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(J\_m \cdot 2\right) \cdot 1}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-U\_m\right) \cdot \cos \left(K \cdot 0.5\right)}{\sqrt{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 -2e+295)
      (- U_m)
      (if (<= t_2 5e+265)
        (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* J_m 2.0) 1.0)) 2.0))))
        (/ (* (- U_m) (cos (* K 0.5))) (sqrt (fma (cos K) 0.5 0.5))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -2e+295) {
		tmp = -U_m;
	} else if (t_2 <= 5e+265) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / ((J_m * 2.0) * 1.0)), 2.0)));
	} else {
		tmp = (-U_m * cos((K * 0.5))) / sqrt(fma(cos(K), 0.5, 0.5));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -2e+295)
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+265)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(J_m * 2.0) * 1.0)) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(-U_m) * cos(Float64(K * 0.5))) / sqrt(fma(cos(K), 0.5, 0.5)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+295], (-U$95$m), If[LessEqual[t$95$2, 5e+265], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(J$95$m * 2.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[((-U$95$m) * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Cos[K], $MachinePrecision] * 0.5 + 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))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.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))))) < 5.0000000000000002e265

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0

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

   5. Applied rewrites 84.1%

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

   if 5.0000000000000002e265 < (*.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 18.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 13.7%

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

   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 50.2

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

   7. Applied rewrites 50.2%

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

   9. Applied rewrites 50.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(J \cdot 2\right) \cdot 1}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-U\right) \cdot \cos \left(K \cdot 0.5\right)}{\sqrt{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \frac{U\_m}{J\_m \cdot 2}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(\left(-2 \cdot J\_m\right) \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_2 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot t\_0\right) \cdot \sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-U\_m\right) \cdot t\_0}{\sqrt{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (/ U_m (* J_m 2.0)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* (* -2.0 J_m) t_2)
          (sqrt (+ 1.0 (pow (/ U_m (* t_2 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_3 -2e+295)
      (- U_m)
      (if (<= t_3 5e+265)
        (* J_m (* (* -2.0 t_0) (sqrt (fma t_1 t_1 1.0))))
        (/ (* (- U_m) t_0) (sqrt (fma (cos K) 0.5 0.5))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double t_1 = U_m / (J_m * 2.0);
	double t_2 = cos((K / 2.0));
	double t_3 = ((-2.0 * J_m) * t_2) * sqrt((1.0 + pow((U_m / (t_2 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_3 <= -2e+295) {
		tmp = -U_m;
	} else if (t_3 <= 5e+265) {
		tmp = J_m * ((-2.0 * t_0) * sqrt(fma(t_1, t_1, 1.0)));
	} else {
		tmp = (-U_m * t_0) / sqrt(fma(cos(K), 0.5, 0.5));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U_m / Float64(J_m * 2.0))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(Float64(-2.0 * J_m) * t_2) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_2 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -2e+295)
		tmp = Float64(-U_m);
	elseif (t_3 <= 5e+265)
		tmp = Float64(J_m * Float64(Float64(-2.0 * t_0) * sqrt(fma(t_1, t_1, 1.0))));
	else
		tmp = Float64(Float64(Float64(-U_m) * t_0) / sqrt(fma(cos(K), 0.5, 0.5)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$2 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$3, -2e+295], (-U$95$m), If[LessEqual[t$95$3, 5e+265], N[(J$95$m * N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-U$95$m) * t$95$0), $MachinePrecision] / N[Sqrt[N[(N[Cos[K], $MachinePrecision] * 0.5 + 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))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.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))))) < 5.0000000000000002e265

   1. Initial program 99.9%

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

   4. Applied rewrites 81.9%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in K around 0

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

      1. lower-*.f64 84.1

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

   9. Applied rewrites 84.1%

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

   if 5.0000000000000002e265 < (*.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 18.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 13.7%

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

   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 50.2

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

   7. Applied rewrites 50.2%

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

   9. Applied rewrites 50.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 2}, \frac{U}{J \cdot 2}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-U\right) \cdot \cos \left(K \cdot 0.5\right)}{\sqrt{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-157}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot t\_0\right) \cdot \sqrt{\mathsf{fma}\left(U\_m, \frac{U\_m}{\left(J\_m \cdot J\_m\right) \cdot 4}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_2 -2e+295)
      (- U_m)
      (if (<= t_2 -1e-157)
        (*
         J_m
         (* (* -2.0 t_0) (sqrt (fma U_m (/ U_m (* (* J_m J_m) 4.0)) 1.0))))
        (* (* -2.0 J_m) t_0))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K * 0.5));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / (t_1 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -2e+295) {
		tmp = -U_m;
	} else if (t_2 <= -1e-157) {
		tmp = J_m * ((-2.0 * t_0) * sqrt(fma(U_m, (U_m / ((J_m * J_m) * 4.0)), 1.0)));
	} else {
		tmp = (-2.0 * J_m) * t_0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K * 0.5))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -2e+295)
		tmp = Float64(-U_m);
	elseif (t_2 <= -1e-157)
		tmp = Float64(J_m * Float64(Float64(-2.0 * t_0) * sqrt(fma(U_m, Float64(U_m / Float64(Float64(J_m * J_m) * 4.0)), 1.0))));
	else
		tmp = Float64(Float64(-2.0 * J_m) * t_0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+295], (-U$95$m), If[LessEqual[t$95$2, -1e-157], N[(J$95$m * N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[Sqrt[N[(U$95$m * N[(U$95$m / N[(N[(J$95$m * J$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$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))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.99999999999999943e-158

   1. Initial program 99.9%

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 80.7

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\left(J \cdot J\right)} \cdot 4}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot J \]

   7. Applied rewrites 80.7%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(U, \frac{U}{\color{blue}{\left(J \cdot J\right) \cdot 4}}, 1\right)} \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot J \]

   if -9.99999999999999943e-158 < (*.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 78.4%

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

   3. Taylor expanded in 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 50.7

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

   5. Applied rewrites 50.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-157}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{\left(J \cdot J\right) \cdot 4}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+270}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U\_m \cdot U\_m}{J\_m}}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(K, K \cdot \left(J\_m \cdot 0.25\right), -2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot \frac{U\_m}{J\_m}}{J\_m}, 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 -2e+270)
      (- U_m)
      (if (<= t_1 -5e-102)
        (* (sqrt (fma 0.25 (/ (/ (* U_m U_m) J_m) J_m) 1.0)) (* -2.0 J_m))
        (*
         (fma K (* K (* J_m 0.25)) (* -2.0 J_m))
         (sqrt (fma 0.25 (/ (* U_m (/ U_m J_m)) J_m) 1.0))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -2e+270) {
		tmp = -U_m;
	} else if (t_1 <= -5e-102) {
		tmp = sqrt(fma(0.25, (((U_m * U_m) / J_m) / J_m), 1.0)) * (-2.0 * J_m);
	} else {
		tmp = fma(K, (K * (J_m * 0.25)), (-2.0 * J_m)) * sqrt(fma(0.25, ((U_m * (U_m / J_m)) / J_m), 1.0));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -2e+270)
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e-102)
		tmp = Float64(sqrt(fma(0.25, Float64(Float64(Float64(U_m * U_m) / J_m) / J_m), 1.0)) * Float64(-2.0 * J_m));
	else
		tmp = Float64(fma(K, Float64(K * Float64(J_m * 0.25)), Float64(-2.0 * J_m)) * sqrt(fma(0.25, Float64(Float64(U_m * Float64(U_m / J_m)) / J_m), 1.0)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+270], (-U$95$m), If[LessEqual[t$95$1, -5e-102], N[(N[Sqrt[N[(0.25 * N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(K * N[(K * N[(J$95$m * 0.25), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * N[(U$95$m / J$95$m), $MachinePrecision]), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $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))))) < -2.0000000000000001e270

   1. Initial program 15.1%

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

   3. Taylor expanded in J around 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 53.7

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

   5. Applied rewrites 53.7%

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

   if -2.0000000000000001e270 < (*.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))))) < -5.00000000000000026e-102

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.6

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

   10. Applied rewrites 53.6%

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

   if -5.00000000000000026e-102 < (*.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 78.9%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 47.3

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 26.9

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

   8. Applied rewrites 26.9%

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

   10. Applied rewrites 35.5%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+270}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U \cdot U}{J}}{J}, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(K, K \cdot \left(J \cdot 0.25\right), -2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot \frac{U}{J}}{J}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+270}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U\_m \cdot U\_m}{J\_m}}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, U\_m \cdot \left(K \cdot K\right), U\_m\right) \cdot \left(-1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 -2e+270)
      (- U_m)
      (if (<= t_1 -2e-240)
        (* (sqrt (fma 0.25 (/ (/ (* U_m U_m) J_m) J_m) 1.0)) (* -2.0 J_m))
        (* (fma -0.125 (* U_m (* K K)) U_m) (- 1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -2e+270) {
		tmp = -U_m;
	} else if (t_1 <= -2e-240) {
		tmp = sqrt(fma(0.25, (((U_m * U_m) / J_m) / J_m), 1.0)) * (-2.0 * J_m);
	} else {
		tmp = fma(-0.125, (U_m * (K * K)), U_m) * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -2e+270)
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-240)
		tmp = Float64(sqrt(fma(0.25, Float64(Float64(Float64(U_m * U_m) / J_m) / J_m), 1.0)) * Float64(-2.0 * J_m));
	else
		tmp = Float64(fma(-0.125, Float64(U_m * Float64(K * K)), U_m) * Float64(-1.0));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+270], (-U$95$m), If[LessEqual[t$95$1, -2e-240], N[(N[Sqrt[N[(0.25 * N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * N[(U$95$m * N[(K * K), $MachinePrecision]), $MachinePrecision] + U$95$m), $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))))) < -2.0000000000000001e270

   1. Initial program 15.1%

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

   3. Taylor expanded in J around 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 53.7

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

   5. Applied rewrites 53.7%

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

   if -2.0000000000000001e270 < (*.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))))) < -1.9999999999999999e-240

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 76.2%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 51.6

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

   10. Applied rewrites 51.6%

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

   if -1.9999999999999999e-240 < (*.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 76.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 73.9%

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

   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 26.6

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

   7. Applied rewrites 26.6%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{neg}\left(1 \cdot \left(U \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 18.2%

       \[\leadsto -1 \cdot \left(U \cdot \cos \left(0.5 \cdot K\right)\right) \]

   11. Taylor expanded in K around 0

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

   13. Applied rewrites 16.4%

       \[\leadsto -1 \cdot \mathsf{fma}\left(-0.125, U \cdot \left(K \cdot K\right), U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+270}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U \cdot U}{J}}{J}, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, U \cdot \left(K \cdot K\right), U\right) \cdot \left(-1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+259}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\left(-2 \cdot J\_m\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J\_m \cdot J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, U\_m \cdot \left(K \cdot K\right), U\_m\right) \cdot \left(-1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))))
   (*
    J_s
    (if (<= t_1 -5e+259)
      (- U_m)
      (if (<= t_1 -1e-128)
        (* (* -2.0 J_m) (sqrt (fma 0.25 (/ (* U_m U_m) (* J_m J_m)) 1.0)))
        (* (fma -0.125 (* U_m (* K K)) U_m) (- 1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -5e+259) {
		tmp = -U_m;
	} else if (t_1 <= -1e-128) {
		tmp = (-2.0 * J_m) * sqrt(fma(0.25, ((U_m * U_m) / (J_m * J_m)), 1.0));
	} else {
		tmp = fma(-0.125, (U_m * (K * K)), U_m) * -1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+259)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-128)
		tmp = Float64(Float64(-2.0 * J_m) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J_m * J_m)), 1.0)));
	else
		tmp = Float64(fma(-0.125, Float64(U_m * Float64(K * K)), U_m) * Float64(-1.0));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+259], (-U$95$m), If[LessEqual[t$95$1, -1e-128], N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J$95$m * J$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * N[(U$95$m * N[(K * K), $MachinePrecision]), $MachinePrecision] + U$95$m), $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))))) < -5.00000000000000033e259

   1. Initial program 23.2%

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

   3. Taylor expanded in 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 50.6

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

   5. Applied rewrites 50.6%

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

   if -5.00000000000000033e259 < (*.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))))) < -1.00000000000000005e-128

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   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 53.5

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

   5. Applied rewrites 53.5%

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

   if -1.00000000000000005e-128 < (*.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 78.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 75.9%

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

   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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 26.6

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

   7. Applied rewrites 26.6%

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

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{neg}\left(1 \cdot \left(U \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 17.7%

       \[\leadsto -1 \cdot \left(U \cdot \cos \left(0.5 \cdot K\right)\right) \]

   11. Taylor expanded in K around 0

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

   13. Applied rewrites 15.6%

       \[\leadsto -1 \cdot \mathsf{fma}\left(-0.125, U \cdot \left(K \cdot K\right), U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+259}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, U \cdot \left(K \cdot K\right), U\right) \cdot \left(-1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \frac{U\_m}{J\_m \cdot 2}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;J\_m \cdot \left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (/ U_m (* J_m 2.0))) (t_1 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J_m 2.0))) 2.0))))
         -2e+295)
      (- U_m)
      (* J_m (* (* -2.0 (cos (* K 0.5))) (sqrt (fma t_0 t_0 1.0))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = U_m / (J_m * 2.0);
	double t_1 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / (t_1 * (J_m * 2.0))), 2.0)))) <= -2e+295) {
		tmp = -U_m;
	} else {
		tmp = J_m * ((-2.0 * cos((K * 0.5))) * sqrt(fma(t_0, t_0, 1.0)));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(U_m / Float64(J_m * 2.0))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J_m * 2.0))) ^ 2.0)))) <= -2e+295)
		tmp = Float64(-U_m);
	else
		tmp = Float64(J_m * Float64(Float64(-2.0 * cos(Float64(K * 0.5))) * sqrt(fma(t_0, t_0, 1.0))));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(U$95$m / N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e+295], (-U$95$m), N[(J$95$m * N[(N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $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))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.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 86.2%

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

   4. Applied rewrites 69.6%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 86.0%

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

   7. Taylor expanded in K around 0

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

      1. lower-*.f64 72.1

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

   9. Applied rewrites 72.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J \cdot 2}, \frac{U}{J \cdot 2}, 1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(J\_m \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U\_m \cdot U\_m}{J\_m}}{J\_m}, 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))
         -2e+295)
      (- U_m)
      (*
       (* -2.0 (* J_m (cos (* K 0.5))))
       (sqrt (fma 0.25 (/ (/ (* U_m U_m) J_m) J_m) 1.0)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))) <= -2e+295) {
		tmp = -U_m;
	} else {
		tmp = (-2.0 * (J_m * cos((K * 0.5)))) * sqrt(fma(0.25, (((U_m * U_m) / J_m) / J_m), 1.0));
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0)))) <= -2e+295)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(-2.0 * Float64(J_m * cos(Float64(K * 0.5)))) * sqrt(fma(0.25, Float64(Float64(Float64(U_m * U_m) / J_m) / J_m), 1.0)));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e+295], (-U$95$m), N[(N[(-2.0 * N[(J$95$m * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $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))))) < -2e295

   1. Initial program 5.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

   if -2e295 < (*.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 86.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 63.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 63.7

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 63.7

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

   9. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot -2\right)} \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U \cdot U}{J}}{J}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{\frac{U \cdot U}{J}}{J}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J\_m \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+270}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(K, K \cdot \left(J\_m \cdot 0.25\right), -2 \cdot J\_m\right) \cdot 1\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J_m 2.0))) 2.0))))
         -2e+270)
      (- U_m)
      (* (fma K (* K (* J_m 0.25)) (* -2.0 J_m)) 1.0)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J_m * 2.0))), 2.0)))) <= -2e+270) {
		tmp = -U_m;
	} else {
		tmp = fma(K, (K * (J_m * 0.25)), (-2.0 * J_m)) * 1.0;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J_m * 2.0))) ^ 2.0)))) <= -2e+270)
		tmp = Float64(-U_m);
	else
		tmp = Float64(fma(K, Float64(K * Float64(J_m * 0.25)), Float64(-2.0 * J_m)) * 1.0);
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e+270], (-U$95$m), N[(N[(K * N[(K * N[(J$95$m * 0.25), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $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))))) < -2.0000000000000001e270

   1. Initial program 15.1%

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

   3. Taylor expanded in J around 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 53.7

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

   5. Applied rewrites 53.7%

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

   if -2.0000000000000001e270 < (*.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 85.9%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 33.2

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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in U around 0

      \[\leadsto \mathsf{fma}\left(K, K \cdot \left(J \cdot \frac{1}{4}\right), J \cdot -2\right) \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 30.9%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+270}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(K, K \cdot \left(J \cdot 0.25\right), -2 \cdot J\right) \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.0% accurate, 124.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \left(-U\_m\right) \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s (- U_m)))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * -U_m;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * -u_m
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * -U_m;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * -U_m

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * Float64(-U_m))
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * -U_m;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * (-U$95$m)), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

3. Taylor expanded in 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 28.3

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

5. Applied rewrites 28.3%

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

Reproduce

herbie shell --seed 2024223 
(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)))))