Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 99.4%

Time: 12.4s

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: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+303}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot t\_2\right)\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{J \cdot \left(2 \cdot t\_2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (cos (* K 0.5))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e+303)
       (*
        (* -2.0 (* J t_2))
        (sqrt (+ 1.0 (pow (/ U_m (* J (* 2.0 t_2))) 2.0))))
       (* (- U_m) (fma -2.0 (/ (* J J) (* U_m U_m)) -1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = cos((K * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+303) {
		tmp = (-2.0 * (J * t_2)) * sqrt((1.0 + pow((U_m / (J * (2.0 * t_2))), 2.0)));
	} else {
		tmp = -U_m * fma(-2.0, ((J * J) / (U_m * U_m)), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+303)
		tmp = Float64(Float64(-2.0 * Float64(J * t_2)) * sqrt(Float64(1.0 + (Float64(U_m / Float64(J * Float64(2.0 * t_2))) ^ 2.0))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J * J) / Float64(U_m * U_m)), -1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+303], N[(N[(-2.0 * N[(J * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1e303

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 1e303 < (*.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 5.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in U around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 53.6

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

   11. Applied rewrites 53.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+246}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-263}:\\ \;\;\;\;J \cdot \left(-2 \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{J} \cdot \frac{U\_m}{J}, 0.25, 1\right)}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+303}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (* (* -2.0 J) (cos (* K 0.5)))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -5e+246)
       t_2
       (if (<= t_1 -5e-263)
         (* J (* -2.0 (sqrt (fma (* (/ U_m J) (/ U_m J)) 0.25 1.0))))
         (if (<= t_1 1e+303)
           t_2
           (* (- U_m) (fma -2.0 (/ (* J J) (* U_m U_m)) -1.0))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = (-2.0 * J) * cos((K * 0.5));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -5e+246) {
		tmp = t_2;
	} else if (t_1 <= -5e-263) {
		tmp = J * (-2.0 * sqrt(fma(((U_m / J) * (U_m / J)), 0.25, 1.0)));
	} else if (t_1 <= 1e+303) {
		tmp = t_2;
	} else {
		tmp = -U_m * fma(-2.0, ((J * J) / (U_m * U_m)), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = Float64(Float64(-2.0 * J) * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -5e+246)
		tmp = t_2;
	elseif (t_1 <= -5e-263)
		tmp = Float64(J * Float64(-2.0 * sqrt(fma(Float64(Float64(U_m / J) * Float64(U_m / J)), 0.25, 1.0))));
	elseif (t_1 <= 1e+303)
		tmp = t_2;
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J * J) / Float64(U_m * U_m)), -1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -5e+246], t$95$2, If[LessEqual[t$95$1, -5e-263], N[(J * N[(-2.0 * N[Sqrt[N[(N[(N[(U$95$m / J), $MachinePrecision] * N[(U$95$m / J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+303], t$95$2, N[((-U$95$m) * N[(-2.0 * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.99999999999999976e246 or -5.00000000000000006e-263 < (*.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))))) < 1e303

   1. Initial program 99.8%

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

   3. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if -4.99999999999999976e246 < (*.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))))) < -5.00000000000000006e-263

   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 0

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

      1. lower-*.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.5

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

   8. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 61.5%

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

   if 1e303 < (*.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 5.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in U around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 53.6

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

   11. Applied rewrites 53.6%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{+246}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-263}:\\ \;\;\;\;J \cdot \left(-2 \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot \frac{U}{J}, 0.25, 1\right)}\right)\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+156}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U\_m \cdot U\_m}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{-150}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -1e+156)
       (* -2.0 J)
       (if (<= t_1 -2e-152)
         (* (* -2.0 J) (sqrt (fma 0.25 (/ (* U_m U_m) (* J J)) 1.0)))
         (if (<= t_1 1e-150)
           (- U_m)
           (* (- U_m) (fma -2.0 (/ (* J J) (* U_m U_m)) -1.0))))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e+156) {
		tmp = -2.0 * J;
	} else if (t_1 <= -2e-152) {
		tmp = (-2.0 * J) * sqrt(fma(0.25, ((U_m * U_m) / (J * J)), 1.0));
	} else if (t_1 <= 1e-150) {
		tmp = -U_m;
	} else {
		tmp = -U_m * fma(-2.0, ((J * J) / (U_m * U_m)), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+156)
		tmp = Float64(-2.0 * J);
	elseif (t_1 <= -2e-152)
		tmp = Float64(Float64(-2.0 * J) * sqrt(fma(0.25, Float64(Float64(U_m * U_m) / Float64(J * J)), 1.0)));
	elseif (t_1 <= 1e-150)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J * J) / Float64(U_m * U_m)), -1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e+156], N[(-2.0 * J), $MachinePrecision], If[LessEqual[t$95$1, -2e-152], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[N[(0.25 * N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-150], (-U$95$m), N[((-U$95$m) * N[(-2.0 * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or -2.00000000000000013e-152 < (*.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))))) < 1.00000000000000001e-150

   1. Initial program 39.6%

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

   3. Taylor expanded in J around 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 39.8

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

   5. Applied rewrites 39.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.9999999999999998e155

   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 0

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

      1. lower-*.f64 43.8

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

   5. Applied rewrites 43.8%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 30.2

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

   8. Applied rewrites 30.2%

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

   if -9.9999999999999998e155 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000013e-152

   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 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 55.7

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

   5. Applied rewrites 55.7%

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

   if 1.00000000000000001e-150 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 76.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 47.9

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

   8. Applied rewrites 47.9%

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

   9. Taylor expanded in U around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 25.9

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

   11. Applied rewrites 25.9%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{+156}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \sqrt{\mathsf{fma}\left(0.25, \frac{U \cdot U}{J \cdot J}, 1\right)}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{-150}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-82}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;t\_2 \leq 10^{+303}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(U\_m, \frac{U\_m}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(J, \cos K, J\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 5e-82)
       (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* J 2.0)) 2.0))))
       (if (<= t_2 1e+303)
         (*
          (* -2.0 (* J (cos (* K 0.5))))
          (sqrt (fma U_m (/ U_m (* (* J 2.0) (fma J (cos K) J))) 1.0)))
         (* (- U_m) (fma -2.0 (/ (* J J) (* U_m U_m)) -1.0)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e-82) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / (J * 2.0)), 2.0)));
	} else if (t_2 <= 1e+303) {
		tmp = (-2.0 * (J * cos((K * 0.5)))) * sqrt(fma(U_m, (U_m / ((J * 2.0) * fma(J, cos(K), J))), 1.0));
	} else {
		tmp = -U_m * fma(-2.0, ((J * J) / (U_m * U_m)), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e-82)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(J * 2.0)) ^ 2.0))));
	elseif (t_2 <= 1e+303)
		tmp = Float64(Float64(-2.0 * Float64(J * cos(Float64(K * 0.5)))) * sqrt(fma(U_m, Float64(U_m / Float64(Float64(J * 2.0) * fma(J, cos(K), J))), 1.0)));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J * J) / Float64(U_m * U_m)), -1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e-82], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+303], N[(N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U$95$m * N[(U$95$m / N[(N[(J * 2.0), $MachinePrecision] * N[(J * N[Cos[K], $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.9999999999999998e-82

   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 0

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

   5. Applied rewrites 85.1%

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

   if 4.9999999999999998e-82 < (*.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))))) < 1e303

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 98.2%

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

   if 1e303 < (*.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 5.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in U around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 53.6

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

   11. Applied rewrites 53.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(U, \frac{U}{\left(J \cdot 2\right) \cdot \mathsf{fma}\left(J, \cos K, J\right)}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;t\_1 \leq 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -1e+18)
       (* -2.0 J)
       (if (<= t_1 1e-150)
         (fma -2.0 (/ (* J J) U_m) (- U_m))
         (* (- U_m) (fma -2.0 (/ (* J J) (* U_m U_m)) -1.0)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e+18) {
		tmp = -2.0 * J;
	} else if (t_1 <= 1e-150) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else {
		tmp = -U_m * fma(-2.0, ((J * J) / (U_m * U_m)), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+18)
		tmp = Float64(-2.0 * J);
	elseif (t_1 <= 1e-150)
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J * J) / Float64(U_m * U_m)), -1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e+18], N[(-2.0 * J), $MachinePrecision], If[LessEqual[t$95$1, 1e-150], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e18

   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 0

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

      1. lower-*.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 35.5

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

   8. Applied rewrites 35.5%

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

   if -1e18 < (*.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))))) < 1.00000000000000001e-150

   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 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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 15.5

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

   5. Applied rewrites 15.5%

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

   6. Taylor expanded in K around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 15.5

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

   8. Applied rewrites 15.5%

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

   if 1.00000000000000001e-150 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 76.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 47.9

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

   8. Applied rewrites 47.9%

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

   9. Taylor expanded in U around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 25.9

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

   11. Applied rewrites 25.9%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-263}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\frac{U\_m}{J} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -1e+18)
       (* -2.0 J)
       (if (<= t_1 -5e-263)
         (fma -2.0 (/ (* J J) U_m) (- U_m))
         (* (* -2.0 J) (* (/ U_m J) -0.5)))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e+18) {
		tmp = -2.0 * J;
	} else if (t_1 <= -5e-263) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else {
		tmp = (-2.0 * J) * ((U_m / J) * -0.5);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+18)
		tmp = Float64(-2.0 * J);
	elseif (t_1 <= -5e-263)
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	else
		tmp = Float64(Float64(-2.0 * J) * Float64(Float64(U_m / J) * -0.5));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e+18], N[(-2.0 * J), $MachinePrecision], If[LessEqual[t$95$1, -5e-263], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], N[(N[(-2.0 * J), $MachinePrecision] * N[(N[(U$95$m / J), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e18

   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 0

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

      1. lower-*.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 35.5

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

   8. Applied rewrites 35.5%

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

   if -1e18 < (*.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))))) < -5.00000000000000006e-263

   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 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. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 18.3

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

   5. Applied rewrites 18.3%

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

   6. Taylor expanded in K around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 18.3

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

   8. Applied rewrites 18.3%

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

   if -5.00000000000000006e-263 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 78.2%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 45.5

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

   5. Applied rewrites 45.5%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.0

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

   8. Applied rewrites 49.0%

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

   9. Taylor expanded in U around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 22.5

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

   11. Applied rewrites 22.5%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -5 \cdot 10^{-263}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U}, -U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\frac{U}{J} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+303}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m}{\mathsf{fma}\left(J, \cos K, J\right)}, \frac{U\_m}{J \cdot 2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e+303)
       (*
        (* -2.0 (* J (cos (* K 0.5))))
        (sqrt (fma (/ U_m (fma J (cos K) J)) (/ U_m (* J 2.0)) 1.0)))
       (* (- U_m) (fma -2.0 (/ (* J J) (* U_m U_m)) -1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+303) {
		tmp = (-2.0 * (J * cos((K * 0.5)))) * sqrt(fma((U_m / fma(J, cos(K), J)), (U_m / (J * 2.0)), 1.0));
	} else {
		tmp = -U_m * fma(-2.0, ((J * J) / (U_m * U_m)), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+303)
		tmp = Float64(Float64(-2.0 * Float64(J * cos(Float64(K * 0.5)))) * sqrt(fma(Float64(U_m / fma(J, cos(K), J)), Float64(U_m / Float64(J * 2.0)), 1.0)));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J * J) / Float64(U_m * U_m)), -1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+303], N[(N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m / N[(J * N[Cos[K], $MachinePrecision] + J), $MachinePrecision]), $MachinePrecision] * N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1e303

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 98.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lower-fma.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if 1e303 < (*.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 5.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in U around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 53.6

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

   11. Applied rewrites 53.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{\mathsf{fma}\left(J, \cos K, J\right)}, \frac{U}{J \cdot 2}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+303}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{J \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 1e+303)
       (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* J 2.0)) 2.0))))
       (* (- U_m) (fma -2.0 (/ (* J J) (* U_m U_m)) -1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+303) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / (J * 2.0)), 2.0)));
	} else {
		tmp = -U_m * fma(-2.0, ((J * J) / (U_m * U_m)), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+303)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(J * 2.0)) ^ 2.0))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J * J) / Float64(U_m * U_m)), -1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+303], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1e303

   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

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

   5. Applied rewrites 86.5%

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

   if 1e303 < (*.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 5.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in U around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 53.6

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

   11. Applied rewrites 53.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{t\_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+303}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{1 + \frac{U\_m \cdot \frac{U\_m}{J \cdot 2}}{J \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m \cdot U\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e+303)
       (*
        (* -2.0 (* J (cos (* K 0.5))))
        (sqrt (+ 1.0 (/ (* U_m (/ U_m (* J 2.0))) (* J 2.0)))))
       (* (- U_m) (fma -2.0 (/ (* J J) (* U_m U_m)) -1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+303) {
		tmp = (-2.0 * (J * cos((K * 0.5)))) * sqrt((1.0 + ((U_m * (U_m / (J * 2.0))) / (J * 2.0))));
	} else {
		tmp = -U_m * fma(-2.0, ((J * J) / (U_m * U_m)), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+303)
		tmp = Float64(Float64(-2.0 * Float64(J * cos(Float64(K * 0.5)))) * sqrt(Float64(1.0 + Float64(Float64(U_m * Float64(U_m / Float64(J * 2.0))) / Float64(J * 2.0)))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J * J) / Float64(U_m * U_m)), -1.0));
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+303], N[(N[(-2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[(U$95$m * N[(U$95$m / N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-U$95$m) * N[(-2.0 * N[(N[(J * J), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1e303

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in K around 0

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

      1. lower-*.f64 86.0

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

   9. Applied rewrites 86.0%

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

   if 1e303 < (*.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 5.6%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 5.0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in U around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 53.6

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

   11. Applied rewrites 53.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;\left(-2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right) \cdot \sqrt{1 + \frac{U \cdot \frac{U}{J \cdot 2}}{J \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-U\right) \cdot \mathsf{fma}\left(-2, \frac{J \cdot J}{U \cdot U}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 0.5× speedup

Math

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

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 1e-150)
       (* J (* -2.0 (sqrt (fma (* (/ U_m J) (/ U_m J)) 0.25 1.0))))
       (* (- U_m) (fma -2.0 (/ (* J J) (* U_m U_m)) -1.0))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e-150) {
		tmp = J * (-2.0 * sqrt(fma(((U_m / J) * (U_m / J)), 0.25, 1.0)));
	} else {
		tmp = -U_m * fma(-2.0, ((J * J) / (U_m * U_m)), -1.0);
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e-150)
		tmp = Float64(J * Float64(-2.0 * sqrt(fma(Float64(Float64(U_m / J) * Float64(U_m / J)), 0.25, 1.0))));
	else
		tmp = Float64(Float64(-U_m) * fma(-2.0, Float64(Float64(J * J) / Float64(U_m * U_m)), -1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.6%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   5. Applied rewrites 50.3%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 1.00000000000000001e-150

   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 0

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

      1. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 57.1

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

   8. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 57.1%

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

   if 1.00000000000000001e-150 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 76.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 47.9

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

   8. Applied rewrites 47.9%

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

   9. Taylor expanded in U around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 25.9

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

   11. Applied rewrites 25.9%

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

4. Final simplification 41.3%

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

Alternative 11: 32.0% accurate, 12.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 1060000000:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{J \cdot J}{U\_m}, -U\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 1060000000.0) (fma -2.0 (/ (* J J) U_m) (- U_m)) (* -2.0 J)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1060000000.0) {
		tmp = fma(-2.0, ((J * J) / U_m), -U_m);
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 1060000000.0)
		tmp = fma(-2.0, Float64(Float64(J * J) / U_m), Float64(-U_m));
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 1060000000.0], N[(-2.0 * N[(N[(J * J), $MachinePrecision] / U$95$m), $MachinePrecision] + (-U$95$m)), $MachinePrecision], N[(-2.0 * J), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.06e9

   1. Initial program 72.1%

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

   3. Taylor expanded in J around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 24.8

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

   5. Applied rewrites 24.8%

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

   6. Taylor expanded in K around 0

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

      1. sub-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 24.8

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

   8. Applied rewrites 24.8%

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

   if 1.06e9 < J

   1. Initial program 98.0%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 40.4

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

   8. Applied rewrites 40.4%

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

Alternative 12: 32.4% accurate, 31.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} \mathbf{if}\;J \leq 1060000000:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (if (<= J 1060000000.0) (- U_m) (* -2.0 J)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1060000000.0) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * J;
	}
	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) :: tmp
    if (j <= 1060000000.0d0) then
        tmp = -u_m
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double tmp;
	if (J <= 1060000000.0) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	tmp = 0
	if J <= 1060000000.0:
		tmp = -U_m
	else:
		tmp = -2.0 * J
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	tmp = 0.0
	if (J <= 1060000000.0)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	tmp = 0.0;
	if (J <= 1060000000.0)
		tmp = -U_m;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := If[LessEqual[J, 1060000000.0], (-U$95$m), N[(-2.0 * J), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.06e9

   1. Initial program 72.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 25.1

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

   5. Applied rewrites 25.1%

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

   if 1.06e9 < J

   1. Initial program 98.0%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in J around inf

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

      1. lower-*.f64 40.4

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

   8. Applied rewrites 40.4%

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

Alternative 13: 26.4% 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 77.7%

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

3. Taylor expanded in J around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 22.9

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

5. Applied rewrites 22.9%

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

Reproduce

herbie shell --seed 2024216 
(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)))))