Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.8% → 98.8%

Time: 10.2s

Alternatives: 13

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: 73.8% 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: 98.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 5e+292) {
		tmp = ((-2.0 * J) * Math.cos((0.5 * K))) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * Math.cos((-0.5 * K)))), 2.0)));
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 5e+292:
		tmp = ((-2.0 * J) * math.cos((0.5 * K))) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * math.cos((-0.5 * K)))), 2.0)))
	else:
		tmp = -1.0 * -U_m
	return tmp

Julia

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

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 5e+292)
		tmp = ((-2.0 * J) * cos((0.5 * K))) * sqrt((1.0 + ((U_m / ((2.0 * J) * cos((-0.5 * K)))) ^ 2.0)));
	else
		tmp = -1.0 * -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999996e292

   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 inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in K around inf

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

      1. metadata-eval N/A

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

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

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

      3. lower-cos.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 4.9999999999999996e292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.5%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 4.1

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

   8. Applied rewrites 4.1%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 54.1%

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

   12. Taylor expanded in J around 0

       \[\leadsto -1 \cdot \left(-U\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 54.1%

       \[\leadsto -1 \cdot \left(-U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J \cdot J}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{0.5 - -0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot K\right)\right)}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -2e-118)
       (* t_1 (sqrt (fma (* U_m (/ U_m (* J J))) 0.25 1.0)))
       (if (<= t_2 -1e-218)
         (fma
          (* (* J J) -2.0)
          (/ (- 0.5 (* -0.5 (cos (* 2.0 (* 0.5 K))))) U_m)
          (- U_m))
         (if (<= t_2 5e+292)
           (* (* (* -2.0 J) (cos (* 0.5 K))) 1.0)
           (* -1.0 (- U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -2e-118) {
		tmp = t_1 * sqrt(fma((U_m * (U_m / (J * J))), 0.25, 1.0));
	} else if (t_2 <= -1e-218) {
		tmp = fma(((J * J) * -2.0), ((0.5 - (-0.5 * cos((2.0 * (0.5 * K))))) / U_m), -U_m);
	} else if (t_2 <= 5e+292) {
		tmp = ((-2.0 * J) * cos((0.5 * K))) * 1.0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -2e-118)
		tmp = Float64(t_1 * sqrt(fma(Float64(U_m * Float64(U_m / Float64(J * J))), 0.25, 1.0)));
	elseif (t_2 <= -1e-218)
		tmp = fma(Float64(Float64(J * J) * -2.0), Float64(Float64(0.5 - Float64(-0.5 * cos(Float64(2.0 * Float64(0.5 * K))))) / U_m), Float64(-U_m));
	elseif (t_2 <= 5e+292)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(0.5 * K))) * 1.0);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -2e-118], N[(t$95$1 * N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-218], N[(N[(N[(J * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[(0.5 - N[(-0.5 * N[Cos[N[(2.0 * N[(0.5 * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$2, 5e+292], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]]]

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 6.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999997e-118

   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 K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.7

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

   8. Applied rewrites 81.7%

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

   if -1.99999999999999997e-118 < (*.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))))) < -1e-218

   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 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 21.1

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

   5. Applied rewrites 21.1%

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

   7. Applied rewrites 21.1%

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

   if -1e-218 < (*.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))))) < 4.9999999999999996e292

   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

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in K around inf

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

      1. metadata-eval N/A

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

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

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

      3. lower-cos.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

   if 4.9999999999999996e292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.5%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 4.1

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

   8. Applied rewrites 4.1%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 54.1%

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

   12. Taylor expanded in J around 0

       \[\leadsto -1 \cdot \left(-U\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 54.1%

       \[\leadsto -1 \cdot \left(-U\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U\_m \cdot \frac{\frac{U\_m}{J}}{J}, 0.25, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{0.5 - -0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot K\right)\right)}{U\_m}, -U\_m\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-118)
       (* (sqrt (fma (* U_m (/ (/ U_m J) J)) 0.25 1.0)) (* -2.0 J))
       (if (<= t_1 -1e-218)
         (fma
          (* (* J J) -2.0)
          (/ (- 0.5 (* -0.5 (cos (* 2.0 (* 0.5 K))))) U_m)
          (- U_m))
         (if (<= t_1 5e+292)
           (* (* (* -2.0 J) (cos (* 0.5 K))) 1.0)
           (* -1.0 (- U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-118) {
		tmp = sqrt(fma((U_m * ((U_m / J) / J)), 0.25, 1.0)) * (-2.0 * J);
	} else if (t_1 <= -1e-218) {
		tmp = fma(((J * J) * -2.0), ((0.5 - (-0.5 * cos((2.0 * (0.5 * K))))) / U_m), -U_m);
	} else if (t_1 <= 5e+292) {
		tmp = ((-2.0 * J) * cos((0.5 * K))) * 1.0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-118)
		tmp = Float64(sqrt(fma(Float64(U_m * Float64(Float64(U_m / J) / J)), 0.25, 1.0)) * Float64(-2.0 * J));
	elseif (t_1 <= -1e-218)
		tmp = fma(Float64(Float64(J * J) * -2.0), Float64(Float64(0.5 - Float64(-0.5 * cos(Float64(2.0 * Float64(0.5 * K))))) / U_m), Float64(-U_m));
	elseif (t_1 <= 5e+292)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(0.5 * K))) * 1.0);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-118], N[(N[Sqrt[N[(N[(U$95$m * N[(N[(U$95$m / J), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-218], N[(N[(N[(J * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[(0.5 - N[(-0.5 * N[Cos[N[(2.0 * N[(0.5 * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], If[LessEqual[t$95$1, 5e+292], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]]

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 6.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999997e-118

   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 K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 53.4

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

   8. Applied rewrites 53.4%

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

   10. Applied rewrites 59.6%

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

   if -1.99999999999999997e-118 < (*.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))))) < -1e-218

   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 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 21.1

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

   5. Applied rewrites 21.1%

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

   7. Applied rewrites 21.1%

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

   if -1e-218 < (*.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))))) < 4.9999999999999996e292

   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

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in K around inf

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

      1. metadata-eval N/A

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

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

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

      3. lower-cos.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

   if 4.9999999999999996e292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.5%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 4.1

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

   8. Applied rewrites 4.1%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 54.1%

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

   12. Taylor expanded in J around 0

       \[\leadsto -1 \cdot \left(-U\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 54.1%

       \[\leadsto -1 \cdot \left(-U\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U\_m \cdot \frac{\frac{U\_m}{J}}{J}, 0.25, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-218}:\\ \;\;\;\;\frac{J \cdot J}{U\_m} \cdot -2 - U\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-118)
       (* (sqrt (fma (* U_m (/ (/ U_m J) J)) 0.25 1.0)) (* -2.0 J))
       (if (<= t_1 -1e-218)
         (- (* (/ (* J J) U_m) -2.0) U_m)
         (if (<= t_1 5e+292)
           (* (* (* -2.0 J) (cos (* 0.5 K))) 1.0)
           (* -1.0 (- U_m))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-118) {
		tmp = sqrt(fma((U_m * ((U_m / J) / J)), 0.25, 1.0)) * (-2.0 * J);
	} else if (t_1 <= -1e-218) {
		tmp = (((J * J) / U_m) * -2.0) - U_m;
	} else if (t_1 <= 5e+292) {
		tmp = ((-2.0 * J) * cos((0.5 * K))) * 1.0;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-118)
		tmp = Float64(sqrt(fma(Float64(U_m * Float64(Float64(U_m / J) / J)), 0.25, 1.0)) * Float64(-2.0 * J));
	elseif (t_1 <= -1e-218)
		tmp = Float64(Float64(Float64(Float64(J * J) / U_m) * -2.0) - U_m);
	elseif (t_1 <= 5e+292)
		tmp = Float64(Float64(Float64(-2.0 * J) * cos(Float64(0.5 * K))) * 1.0);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-118], N[(N[Sqrt[N[(N[(U$95$m * N[(N[(U$95$m / J), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-218], N[(N[(N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] * -2.0), $MachinePrecision] - U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e+292], N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]]

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 6.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999997e-118

   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 K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 53.4

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

   8. Applied rewrites 53.4%

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

   10. Applied rewrites 59.6%

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

   if -1.99999999999999997e-118 < (*.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))))) < -1e-218

   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 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 21.1

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

   5. Applied rewrites 21.1%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 21.1%

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

   if -1e-218 < (*.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))))) < 4.9999999999999996e292

   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

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in K around inf

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

      1. metadata-eval N/A

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

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

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

      3. lower-cos.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

   if 4.9999999999999996e292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.5%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 4.1

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

   8. Applied rewrites 4.1%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 54.1%

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

   12. Taylor expanded in J around 0

       \[\leadsto -1 \cdot \left(-U\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 54.1%

       \[\leadsto -1 \cdot \left(-U\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 60.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U\_m \cdot \frac{\frac{U\_m}{J}}{J}, 0.25, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-289}:\\ \;\;\;\;\frac{J \cdot J}{U\_m} \cdot -2 - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{J}{U\_m} \cdot J}{U\_m} \cdot -2 - 1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-118)
       (* (sqrt (fma (* U_m (/ (/ U_m J) J)) 0.25 1.0)) (* -2.0 J))
       (if (<= t_1 -5e-289)
         (- (* (/ (* J J) U_m) -2.0) U_m)
         (* (- (* (/ (* (/ J U_m) J) U_m) -2.0) 1.0) (- U_m)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-118) {
		tmp = sqrt(fma((U_m * ((U_m / J) / J)), 0.25, 1.0)) * (-2.0 * J);
	} else if (t_1 <= -5e-289) {
		tmp = (((J * J) / U_m) * -2.0) - U_m;
	} else {
		tmp = (((((J / U_m) * J) / U_m) * -2.0) - 1.0) * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-118)
		tmp = Float64(sqrt(fma(Float64(U_m * Float64(Float64(U_m / J) / J)), 0.25, 1.0)) * Float64(-2.0 * J));
	elseif (t_1 <= -5e-289)
		tmp = Float64(Float64(Float64(Float64(J * J) / U_m) * -2.0) - U_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(J / U_m) * J) / U_m) * -2.0) - 1.0) * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-118], N[(N[Sqrt[N[(N[(U$95$m * N[(N[(U$95$m / J), $MachinePrecision] / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-289], N[(N[(N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] * -2.0), $MachinePrecision] - U$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(J / U$95$m), $MachinePrecision] * J), $MachinePrecision] / U$95$m), $MachinePrecision] * -2.0), $MachinePrecision] - 1.0), $MachinePrecision] * (-U$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 6.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999997e-118

   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 K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 53.4

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

   8. Applied rewrites 53.4%

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

   10. Applied rewrites 59.6%

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

   if -1.99999999999999997e-118 < (*.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))))) < -5.00000000000000029e-289

   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 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 18.1%

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

   if -5.00000000000000029e-289 < (*.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 68.4%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 31.6

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 31.7%

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

   13. Applied rewrites 31.7%

       \[\leadsto \left(\frac{\frac{J}{U} \cdot J}{U} \cdot -2 - 1\right) \cdot \left(-U\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 58.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+285}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J \cdot J}, 0.25, 1\right)} \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-289}:\\ \;\;\;\;\frac{J \cdot J}{U\_m} \cdot -2 - U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{J}{U\_m} \cdot J}{U\_m} \cdot -2 - 1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 -5e+285)
     (- U_m)
     (if (<= t_1 -2e-118)
       (* (sqrt (fma (* U_m (/ U_m (* J J))) 0.25 1.0)) (* -2.0 J))
       (if (<= t_1 -5e-289)
         (- (* (/ (* J J) U_m) -2.0) U_m)
         (* (- (* (/ (* (/ J U_m) J) U_m) -2.0) 1.0) (- U_m)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -5e+285) {
		tmp = -U_m;
	} else if (t_1 <= -2e-118) {
		tmp = sqrt(fma((U_m * (U_m / (J * J))), 0.25, 1.0)) * (-2.0 * J);
	} else if (t_1 <= -5e-289) {
		tmp = (((J * J) / U_m) * -2.0) - U_m;
	} else {
		tmp = (((((J / U_m) * J) / U_m) * -2.0) - 1.0) * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+285)
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-118)
		tmp = Float64(sqrt(fma(Float64(U_m * Float64(U_m / Float64(J * J))), 0.25, 1.0)) * Float64(-2.0 * J));
	elseif (t_1 <= -5e-289)
		tmp = Float64(Float64(Float64(Float64(J * J) / U_m) * -2.0) - U_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(J / U_m) * J) / U_m) * -2.0) - 1.0) * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+285], (-U$95$m), If[LessEqual[t$95$1, -2e-118], N[(N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-289], N[(N[(N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] * -2.0), $MachinePrecision] - U$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(J / U$95$m), $MachinePrecision] * J), $MachinePrecision] / U$95$m), $MachinePrecision] * -2.0), $MachinePrecision] - 1.0), $MachinePrecision] * (-U$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000016e285

   1. Initial program 15.2%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.9

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

   5. Applied rewrites 46.9%

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

   if -5.00000000000000016e285 < (*.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))))) < -1.99999999999999997e-118

   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 K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 55.5

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

   8. Applied rewrites 55.5%

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

   if -1.99999999999999997e-118 < (*.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))))) < -5.00000000000000029e-289

   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 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. cos-neg-rev N/A

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

      13. lower-cos.f64 N/A

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

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

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 18.2

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

   5. Applied rewrites 18.2%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 18.1%

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

   if -5.00000000000000029e-289 < (*.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 68.4%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 31.6

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 31.7%

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

   13. Applied rewrites 31.7%

       \[\leadsto \left(\frac{\frac{J}{U} \cdot J}{U} \cdot -2 - 1\right) \cdot \left(-U\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+285}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J}, -0.25, -2 \cdot J\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-289}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{J}{U\_m} \cdot J}{U\_m} \cdot -2 - 1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 -5e+285)
     (- U_m)
     (if (<= t_1 -2e-102)
       (fma (* U_m (/ U_m J)) -0.25 (* -2.0 J))
       (if (<= t_1 -5e-289)
         (- U_m)
         (* (- (* (/ (* (/ J U_m) J) U_m) -2.0) 1.0) (- U_m)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -5e+285) {
		tmp = -U_m;
	} else if (t_1 <= -2e-102) {
		tmp = fma((U_m * (U_m / J)), -0.25, (-2.0 * J));
	} else if (t_1 <= -5e-289) {
		tmp = -U_m;
	} else {
		tmp = (((((J / U_m) * J) / U_m) * -2.0) - 1.0) * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+285)
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-102)
		tmp = fma(Float64(U_m * Float64(U_m / J)), -0.25, Float64(-2.0 * J));
	elseif (t_1 <= -5e-289)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(J / U_m) * J) / U_m) * -2.0) - 1.0) * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+285], (-U$95$m), If[LessEqual[t$95$1, -2e-102], N[(N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * -0.25 + N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-289], (-U$95$m), N[(N[(N[(N[(N[(N[(J / U$95$m), $MachinePrecision] * J), $MachinePrecision] / U$95$m), $MachinePrecision] * -2.0), $MachinePrecision] - 1.0), $MachinePrecision] * (-U$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000016e285 or -1.99999999999999987e-102 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000029e-289

   1. Initial program 41.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 37.3

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

   5. Applied rewrites 37.3%

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

   if -5.00000000000000016e285 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999987e-102

   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 K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 56.2

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

   8. Applied rewrites 56.2%

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

   9. Taylor expanded in U around 0

      \[\leadsto -2 \cdot J + \color{blue}{\frac{-1}{4} \cdot \frac{{U}^{2}}{J}} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.3%

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

   if -5.00000000000000029e-289 < (*.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 68.4%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 31.6

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 31.7%

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

   13. Applied rewrites 31.7%

       \[\leadsto \left(\frac{\frac{J}{U} \cdot J}{U} \cdot -2 - 1\right) \cdot \left(-U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+285}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J}, -0.25, -2 \cdot J\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-289}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 -5e+285)
     (- U_m)
     (if (<= t_1 -2e-102)
       (fma (* U_m (/ U_m J)) -0.25 (* -2.0 J))
       (if (<= t_1 -5e-289) (- U_m) (* -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 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -5e+285) {
		tmp = -U_m;
	} else if (t_1 <= -2e-102) {
		tmp = fma((U_m * (U_m / J)), -0.25, (-2.0 * J));
	} else if (t_1 <= -5e-289) {
		tmp = -U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+285)
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-102)
		tmp = fma(Float64(U_m * Float64(U_m / J)), -0.25, Float64(-2.0 * J));
	elseif (t_1 <= -5e-289)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+285], (-U$95$m), If[LessEqual[t$95$1, -2e-102], N[(N[(U$95$m * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * -0.25 + N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-289], (-U$95$m), N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000016e285 or -1.99999999999999987e-102 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000029e-289

   1. Initial program 41.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 37.3

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

   5. Applied rewrites 37.3%

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

   if -5.00000000000000016e285 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999987e-102

   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 K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 56.2

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

   8. Applied rewrites 56.2%

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

   9. Taylor expanded in U around 0

      \[\leadsto -2 \cdot J + \color{blue}{\frac{-1}{4} \cdot \frac{{U}^{2}}{J}} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.3%

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

   if -5.00000000000000029e-289 < (*.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 68.4%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 31.6

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 31.7%

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

   12. Taylor expanded in J around 0

       \[\leadsto -1 \cdot \left(-U\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 31.9%

       \[\leadsto -1 \cdot \left(-U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+285}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-102}:\\ \;\;\;\;1 \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-289}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 -5e+285)
     (- U_m)
     (if (<= t_1 -2e-102)
       (* 1.0 (* -2.0 J))
       (if (<= t_1 -5e-289) (- U_m) (* -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 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -5e+285) {
		tmp = -U_m;
	} else if (t_1 <= -2e-102) {
		tmp = 1.0 * (-2.0 * J);
	} else if (t_1 <= -5e-289) {
		tmp = -U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j) * t_0)) ** 2.0d0)))
    if (t_1 <= (-5d+285)) then
        tmp = -u_m
    else if (t_1 <= (-2d-102)) then
        tmp = 1.0d0 * ((-2.0d0) * j)
    else if (t_1 <= (-5d-289)) then
        tmp = -u_m
    else
        tmp = (-1.0d0) * -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -5e+285) {
		tmp = -U_m;
	} else if (t_1 <= -2e-102) {
		tmp = 1.0 * (-2.0 * J);
	} else if (t_1 <= -5e-289) {
		tmp = -U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -5e+285:
		tmp = -U_m
	elif t_1 <= -2e-102:
		tmp = 1.0 * (-2.0 * J)
	elif t_1 <= -5e-289:
		tmp = -U_m
	else:
		tmp = -1.0 * -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -5e+285)
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-102)
		tmp = Float64(1.0 * Float64(-2.0 * J));
	elseif (t_1 <= -5e-289)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -5e+285)
		tmp = -U_m;
	elseif (t_1 <= -2e-102)
		tmp = 1.0 * (-2.0 * J);
	elseif (t_1 <= -5e-289)
		tmp = -U_m;
	else
		tmp = -1.0 * -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+285], (-U$95$m), If[LessEqual[t$95$1, -2e-102], N[(1.0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-289], (-U$95$m), N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000016e285 or -1.99999999999999987e-102 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000029e-289

   1. Initial program 41.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 37.3

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

   5. Applied rewrites 37.3%

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

   if -5.00000000000000016e285 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.99999999999999987e-102

   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 K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 56.2

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

   8. Applied rewrites 56.2%

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

   9. Taylor expanded in J around inf

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

   11. Applied rewrites 41.2%

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

   if -5.00000000000000029e-289 < (*.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 68.4%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 31.6

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 31.7%

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

   12. Taylor expanded in J around 0

       \[\leadsto -1 \cdot \left(-U\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 31.9%

       \[\leadsto -1 \cdot \left(-U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 88.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 5e+292)
       (*
        t_1
        (sqrt
         (+ 1.0 (pow (/ U_m (* (* 2.0 J) (fma -0.125 (* K K) 1.0))) 2.0))))
       (* -1.0 (- U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+292) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * fma(-0.125, (K * K), 1.0))), 2.0)));
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+292)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * fma(-0.125, Float64(K * K), 1.0))) ^ 2.0))));
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+292], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * (-U$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999996e292

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if 4.9999999999999996e292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.5%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 4.1

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

   8. Applied rewrites 4.1%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 54.1%

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

   12. Taylor expanded in J around 0

       \[\leadsto -1 \cdot \left(-U\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 54.1%

       \[\leadsto -1 \cdot \left(-U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 88.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999996e292

   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 inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.5

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

   8. Applied rewrites 71.5%

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

   10. Applied rewrites 78.6%

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

   if 4.9999999999999996e292 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 5.5%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 4.1

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

   8. Applied rewrites 4.1%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 54.1%

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

   12. Taylor expanded in J around 0

       \[\leadsto -1 \cdot \left(-U\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 54.1%

       \[\leadsto -1 \cdot \left(-U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 51.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \leq -5 \cdot 10^{-289}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-U\_m\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<=
        (*
         (* (* -2.0 J) t_0)
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))
        -5e-289)
     (- U_m)
     (* -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 tmp;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)))) <= -5e-289) {
		tmp = -U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Fortran

U_m = abs(u)
real(8) function code(j, k, u_m)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j) * t_0)) ** 2.0d0)))) <= (-5d-289)) then
        tmp = -u_m
    else
        tmp = (-1.0d0) * -u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))) <= -5e-289) {
		tmp = -U_m;
	} else {
		tmp = -1.0 * -U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))) <= -5e-289:
		tmp = -U_m
	else:
		tmp = -1.0 * -U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0)))) <= -5e-289)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-1.0 * Float64(-U_m));
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)))) <= -5e-289)
		tmp = -U_m;
	else
		tmp = -1.0 * -U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-289], (-U$95$m), N[(-1.0 * (-U$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000029e-289

   1. Initial program 74.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 23.6

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

   5. Applied rewrites 23.6%

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

   if -5.00000000000000029e-289 < (*.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 68.4%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 31.6

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

   8. Applied rewrites 31.6%

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

   9. Taylor expanded in U around -inf

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

   11. Applied rewrites 31.7%

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

   12. Taylor expanded in J around 0

       \[\leadsto -1 \cdot \left(-U\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 31.9%

       \[\leadsto -1 \cdot \left(-U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 26.2% accurate, 124.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

3. Taylor expanded in J around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 26.9

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

5. Applied rewrites 26.9%

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

Reproduce

herbie shell --seed 2024340 
(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)))))