Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.5% → 99.8%

Time: 11.6s

Alternatives: 12

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (fma -2.0 (/ (* J J) U_m) (- U_m))
     (if (<= t_1 1e+308) t_1 U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	elseif (t_1 <= 1e+308)
		tmp = t_1;
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], t$95$1, U$95$m]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in J around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in K around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   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))))) < 1e308

   1. Initial program 99.8%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.9%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 47.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+308}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := t\_0 \cdot \left(-2 \cdot J\right)\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_3 := \sqrt{1 + {\left(\frac{U\_m}{J \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{+161}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-147}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+308}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* t_0 (* -2.0 J)))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_3 (* (sqrt (+ 1.0 (pow (/ U_m (* J 2.0)) 2.0))) (* -2.0 J))))
   (if (<= t_2 (- INFINITY))
     (fma -2.0 (/ (* J J) U_m) (- U_m))
     (if (<= t_2 -4e+161)
       t_3
       (if (<= t_2 -5e-147)
         (* t_1 (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
         (if (<= t_2 -5e-292)
           t_3
           (if (<= t_2 1e+308) (* (* -2.0 J) (cos (* K 0.5))) U_m)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = t_0 * (-2.0 * J);
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_3 = sqrt((1.0 + pow((U_m / (J * 2.0)), 2.0))) * (-2.0 * J);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else if (t_2 <= -4e+161) {
		tmp = t_3;
	} else if (t_2 <= -5e-147) {
		tmp = t_1 * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else if (t_2 <= -5e-292) {
		tmp = t_3;
	} else if (t_2 <= 1e+308) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(t_0 * Float64(-2.0 * J))
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_3 = Float64(sqrt(Float64(1.0 + (Float64(U_m / Float64(J * 2.0)) ^ 2.0))) * Float64(-2.0 * J))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	elseif (t_2 <= -4e+161)
		tmp = t_3;
	elseif (t_2 <= -5e-147)
		tmp = Float64(t_1 * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J * J)), 1.0)));
	elseif (t_2 <= -5e-292)
		tmp = t_3;
	elseif (t_2 <= 1e+308)
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, -4e+161], t$95$3, If[LessEqual[t$95$2, -5e-147], N[(t$95$1 * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-292], t$95$3, If[LessEqual[t$95$2, 1e+308], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in J around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in K around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000002e161 or -5.00000000000000013e-147 < (*.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))))) < -4.99999999999999981e-292

   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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 88.4

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

   5. Simplified 88.4%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 58.9

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

   8. Simplified 58.9%

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

   if -4.0000000000000002e161 < (*.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))))) < -5.00000000000000013e-147

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. sqrt-lowering-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. accelerator-lowering-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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.f64 95.3

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

   5. Simplified 95.3%

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

   if -4.99999999999999981e-292 < (*.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))))) < 1e308

   1. Initial program 99.7%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 67.4

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

   5. Simplified 67.4%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.9%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 47.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -4 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-147}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+308}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+150}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;t\_1 \leq -3 \cdot 10^{-132}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (fma -2.0 (/ (* J J) U_m) (- U_m))
     (if (<= t_1 -2e+150)
       (* -2.0 J)
       (if (<= t_1 -3e-132)
         (* (* -2.0 J) (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
         (if (<= t_1 -5e-292) (- U_m) U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else if (t_1 <= -2e+150) {
		tmp = -2.0 * J;
	} else if (t_1 <= -3e-132) {
		tmp = (-2.0 * J) * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else if (t_1 <= -5e-292) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	elseif (t_1 <= -2e+150)
		tmp = Float64(-2.0 * J);
	elseif (t_1 <= -3e-132)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J * J)), 1.0)));
	elseif (t_1 <= -5e-292)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, -2e+150], N[(-2.0 * J), $MachinePrecision], If[LessEqual[t$95$1, -3e-132], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-292], (-U$95$m), U$95$m]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in J around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in K around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999996e150

   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 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. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 79.5

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

   5. Simplified 79.5%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 41.3

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

   8. Simplified 41.3%

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

   if -1.99999999999999996e150 < (*.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))))) < -3e-132

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-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. accelerator-lowering-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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 69.1

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

   5. Simplified 69.1%

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

   if -3e-132 < (*.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))))) < -4.99999999999999981e-292

   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 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. neg-lowering-neg.f64 21.6

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

   5. Simplified 21.6%

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

   if -4.99999999999999981e-292 < (*.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 76.7%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 26.3%

      \[\leadsto \color{blue}{U} \]
3. Recombined 5 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{+150}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -3 \cdot 10^{-132}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U\_m}{J \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (fma -2.0 (/ (* J J) U_m) (- U_m))
     (if (<= t_1 -5e-292)
       (* (sqrt (+ 1.0 (pow (/ U_m (* J 2.0)) 2.0))) (* -2.0 J))
       (if (<= t_1 1e+308) (* (* -2.0 J) (cos (* K 0.5))) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else if (t_1 <= -5e-292) {
		tmp = sqrt((1.0 + pow((U_m / (J * 2.0)), 2.0))) * (-2.0 * J);
	} else if (t_1 <= 1e+308) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	elseif (t_1 <= -5e-292)
		tmp = Float64(sqrt(Float64(1.0 + (Float64(U_m / Float64(J * 2.0)) ^ 2.0))) * Float64(-2.0 * J));
	elseif (t_1 <= 1e+308)
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, -5e-292], N[(N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in J around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in K around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999981e-292

   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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 92.1

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

   5. Simplified 92.1%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 63.2

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

   8. Simplified 63.2%

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

   if -4.99999999999999981e-292 < (*.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))))) < 1e308

   1. Initial program 99.7%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 67.4

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

   5. Simplified 67.4%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.9%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 47.1%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+308}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{U\_m}{J \cdot 2}}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(t\_1 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{t\_0}{J}, \frac{t\_0}{\frac{2}{U\_m}}, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+308}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ U_m (* J 2.0))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* t_1 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (fma -2.0 (/ (* J J) U_m) (- U_m))
     (if (<= t_2 -5e-292)
       (* (* -2.0 J) (sqrt (fma (/ t_0 J) (/ t_0 (/ 2.0 U_m)) 1.0)))
       (if (<= t_2 1e+308) (* (* -2.0 J) (cos (* K 0.5))) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = sqrt((U_m / (J * 2.0)));
	double t_1 = cos((K / 2.0));
	double t_2 = (t_1 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else if (t_2 <= -5e-292) {
		tmp = (-2.0 * J) * sqrt(fma((t_0 / J), (t_0 / (2.0 / U_m)), 1.0));
	} else if (t_2 <= 1e+308) {
		tmp = (-2.0 * J) * cos((K * 0.5));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = sqrt(Float64(U_m / Float64(J * 2.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(t_1 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	elseif (t_2 <= -5e-292)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(Float64(t_0 / J), Float64(t_0 / Float64(2.0 / U_m)), 1.0)));
	elseif (t_2 <= 1e+308)
		tmp = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Sqrt[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, -5e-292], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(N[(t$95$0 / J), $MachinePrecision] * N[(t$95$0 / N[(2.0 / U$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+308], N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in J around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in K around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999981e-292

   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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 92.1

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

   5. Simplified 92.1%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. associate-/l* N/A

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

      8. times-frac N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 54.6%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 38.1

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

   10. Simplified 38.1%

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

   if -4.99999999999999981e-292 < (*.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))))) < 1e308

   1. Initial program 99.7%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 67.4

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

   5. Simplified 67.4%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.9%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 47.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{\sqrt{\frac{U}{J \cdot 2}}}{J}, \frac{\sqrt{\frac{U}{J \cdot 2}}}{\frac{2}{U}}, 1\right)}\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+308}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq -3 \cdot 10^{-132}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (fma -2.0 (/ (* J J) U_m) (- U_m))
     (if (<= t_1 -3e-132) (* -2.0 J) (if (<= t_1 -5e-292) (- U_m) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else if (t_1 <= -3e-132) {
		tmp = -2.0 * J;
	} else if (t_1 <= -5e-292) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	elseif (t_1 <= -3e-132)
		tmp = Float64(-2.0 * J);
	elseif (t_1 <= -5e-292)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, -3e-132], N[(-2.0 * J), $MachinePrecision], If[LessEqual[t$95$1, -5e-292], (-U$95$m), U$95$m]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in J around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in K around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   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))))) < -3e-132

   1. Initial program 99.8%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 83.1

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

   5. Simplified 83.1%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 49.1

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

   8. Simplified 49.1%

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

   if -3e-132 < (*.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))))) < -4.99999999999999981e-292

   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 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. neg-lowering-neg.f64 21.6

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

   5. Simplified 21.6%

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

   if -4.99999999999999981e-292 < (*.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 76.7%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 26.3%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -3 \cdot 10^{-132}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -3 \cdot 10^{-132}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -3e-132) (* -2.0 J) (if (<= t_1 -5e-292) (- U_m) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -3e-132) {
		tmp = -2.0 * J;
	} else if (t_1 <= -5e-292) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= -3e-132) {
		tmp = -2.0 * J;
	} else if (t_1 <= -5e-292) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = (t_0 * (-2.0 * J)) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= -3e-132:
		tmp = -2.0 * J
	elif t_1 <= -5e-292:
		tmp = -U_m
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -3e-132)
		tmp = Float64(-2.0 * J);
	elseif (t_1 <= -5e-292)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= -3e-132)
		tmp = -2.0 * J;
	elseif (t_1 <= -5e-292)
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -3e-132], N[(-2.0 * J), $MachinePrecision], If[LessEqual[t$95$1, -5e-292], (-U$95$m), U$95$m]]]]]

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 or -3e-132 < (*.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))))) < -4.99999999999999981e-292

   1. Initial program 30.3%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 39.9

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

   5. Simplified 39.9%

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

   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))))) < -3e-132

   1. Initial program 99.8%

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

   3. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 83.1

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

   5. Simplified 83.1%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 49.1

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

   8. Simplified 49.1%

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

   if -4.99999999999999981e-292 < (*.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 76.7%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 26.3%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -3 \cdot 10^{-132}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := t\_0 \cdot \left(-2 \cdot J\right)\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+308}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{J \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* t_0 (* -2.0 J)))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (fma -2.0 (/ (* J J) U_m) (- U_m))
     (if (<= t_2 1e+308)
       (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* J 2.0)) 2.0))))
       U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = t_0 * (-2.0 * J);
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else if (t_2 <= 1e+308) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / (J * 2.0)), 2.0)));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(t_0 * Float64(-2.0 * J))
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	elseif (t_2 <= 1e+308)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(J * 2.0)) ^ 2.0))));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, 1e+308], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in J around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in K around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   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))))) < 1e308

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 90.2

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

   5. Simplified 90.2%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.9%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 47.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+308}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := U\_m \cdot \frac{0.5}{J}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (* U_m (/ 0.5 J))))
   (if (<= t_1 (- INFINITY))
     (fma -2.0 (/ (* J J) U_m) (- U_m))
     (if (<= t_1 1e+308)
       (/
        1.0
        (/ (/ 1.0 (* (* -2.0 J) (cos (* K 0.5)))) (sqrt (fma t_2 t_2 1.0))))
       U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = U_m * (0.5 / J);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else if (t_1 <= 1e+308) {
		tmp = 1.0 / ((1.0 / ((-2.0 * J) * cos((K * 0.5)))) / sqrt(fma(t_2, t_2, 1.0)));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = Float64(U_m * Float64(0.5 / J))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	elseif (t_1 <= 1e+308)
		tmp = Float64(1.0 / Float64(Float64(1.0 / Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))) / sqrt(fma(t_2, t_2, 1.0))));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U$95$m * N[(0.5 / J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], N[(1.0 / N[(N[(1.0 / N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in J around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in K around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   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))))) < 1e308

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 90.2

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

   5. Simplified 90.2%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. associate-/l* N/A

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

      8. times-frac N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 50.6%

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

   9. Applied egg-rr 50.5%

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

       1. associate-*l/ N/A

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

       2. clear-num N/A

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

       3. /-lowering-/.f64 N/A

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

       4. *-lft-identity N/A

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

       5. /-lowering-/.f64 N/A

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

   11. Applied egg-rr 89.9%

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

   if 1e308 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.9%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 47.1%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+308}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(U \cdot \frac{0.5}{J}, U \cdot \frac{0.5}{J}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{U\_m}{J \cdot 2}}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(t\_1 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_1 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{t\_0}{J}, \frac{t\_0}{\frac{2}{U\_m}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ U_m (* J 2.0))))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* t_1 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U_m (* t_1 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (fma -2.0 (/ (* J J) U_m) (- U_m))
     (if (<= t_2 -5e-292)
       (* (* -2.0 J) (sqrt (fma (/ t_0 J) (/ t_0 (/ 2.0 U_m)) 1.0)))
       U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = sqrt((U_m / (J * 2.0)));
	double t_1 = cos((K / 2.0));
	double t_2 = (t_1 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_1 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else if (t_2 <= -5e-292) {
		tmp = (-2.0 * J) * sqrt(fma((t_0 / J), (t_0 / (2.0 / U_m)), 1.0));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = sqrt(Float64(U_m / Float64(J * 2.0)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(t_1 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_1 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	elseif (t_2 <= -5e-292)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(Float64(t_0 / J), Float64(t_0 / Float64(2.0 / U_m)), 1.0)));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Sqrt[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, -5e-292], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(N[(t$95$0 / J), $MachinePrecision] * N[(t$95$0 / N[(2.0 / U$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   3. Taylor expanded in J around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in K around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.1

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

   8. Simplified 46.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999981e-292

   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

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 92.1

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

   5. Simplified 92.1%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. unpow1 N/A

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

      6. sqr-pow N/A

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

      7. associate-/l* N/A

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

      8. times-frac N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 54.6%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 38.1

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

   10. Simplified 38.1%

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

   if -4.99999999999999981e-292 < (*.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 76.7%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 26.3%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(\frac{\sqrt{\frac{U}{J \cdot 2}}}{J}, \frac{\sqrt{\frac{U}{J \cdot 2}}}{\frac{2}{U}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\left(t\_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<=
        (*
         (* t_0 (* -2.0 J))
         (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))
        -5e-292)
     (- U_m)
     U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (((t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)))) <= -5e-292) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((t_0 * ((-2.0d0) * j)) * sqrt((1.0d0 + ((u_m / (t_0 * (j * 2.0d0))) ** 2.0d0)))) <= (-5d-292)) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (((t_0 * (-2.0 * J)) * Math.sqrt((1.0 + Math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))) <= -5e-292) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if ((t_0 * (-2.0 * J)) * math.sqrt((1.0 + math.pow((U_m / (t_0 * (J * 2.0))), 2.0)))) <= -5e-292:
		tmp = -U_m
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0)))) <= -5e-292)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (((t_0 * (-2.0 * J)) * sqrt((1.0 + ((U_m / (t_0 * (J * 2.0))) ^ 2.0)))) <= -5e-292)
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-292], (-U$95$m), U$95$m]]

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))))) < -4.99999999999999981e-292

   1. Initial program 75.4%

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

   3. Taylor expanded in J around 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. neg-lowering-neg.f64 21.1

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

   5. Simplified 21.1%

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

   if -4.99999999999999981e-292 < (*.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 76.7%

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

   3. Taylor expanded in U around -inf

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

   5. Simplified 26.3%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-292}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.1% accurate, 373.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Taylor expanded in U around -inf

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

5. Simplified 25.7%

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

Reproduce

herbie shell --seed 2024198 
(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)))))