Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.5% → 99.0%

Time: 8.3s

Alternatives: 11

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+289}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\frac{\frac{U\_m}{J\_m}}{-2}}{t\_1}\right)}^{2} + 1} \cdot \left(t\_1 \cdot -2\right)\right) \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m \cdot U\_m} \cdot J\_m, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (cos (* -0.5 K)))
        (t_2
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 1e+289)
        (*
         (* (sqrt (+ (pow (/ (/ (/ U_m J_m) -2.0) t_1) 2.0) 1.0)) (* t_1 -2.0))
         J_m)
        (* (fma (* (/ J_m (* U_m U_m)) J_m) -2.0 -1.0) (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = cos((-0.5 * K));
	double t_2 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+289) {
		tmp = (sqrt((pow((((U_m / J_m) / -2.0) / t_1), 2.0) + 1.0)) * (t_1 * -2.0)) * J_m;
	} else {
		tmp = fma(((J_m / (U_m * U_m)) * J_m), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+289)
		tmp = Float64(Float64(sqrt(Float64((Float64(Float64(Float64(U_m / J_m) / -2.0) / t_1) ^ 2.0) + 1.0)) * Float64(t_1 * -2.0)) * J_m);
	else
		tmp = Float64(fma(Float64(Float64(J_m / Float64(U_m * U_m)) * J_m), -2.0, -1.0) * Float64(-U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+289], N[(N[(N[Sqrt[N[(N[Power[N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] / -2.0), $MachinePrecision] / t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision], N[(N[(N[(N[(J$95$m / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $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 45.1

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

   5. Applied rewrites 45.1%

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

   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))))) < 1.0000000000000001e289

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   if 1.0000000000000001e289 < (*.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 30.5%

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

   3. Taylor expanded in K around 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. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in U around -inf

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

   8. Applied rewrites 34.4%

      \[\leadsto \left(\left(\frac{J}{U \cdot U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(\color{blue}{-2} \cdot J\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 34.4%

       \[\leadsto \left(\left(\frac{-1}{U} \cdot \frac{J}{-U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(-2 \cdot J\right) \]

   11. Taylor expanded in U around -inf

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

   13. Applied rewrites 51.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+289}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\frac{\frac{U}{J}}{-2}}{\cos \left(-0.5 \cdot K\right)}\right)}^{2} + 1} \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot -2\right)\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U \cdot U} \cdot J, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ t_2 := \left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\_m\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+212}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(J\_m \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+180}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m \cdot U\_m} \cdot J\_m, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0))))
        (t_2 (* (* (cos (* -0.5 K)) -2.0) J_m)))
   (*
    J_s
    (if (<= t_1 -5e+307)
      (- U_m)
      (if (<= t_1 -1e+212)
        t_2
        (if (<= t_1 -4e-288)
          (* (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)) (* J_m -2.0))
          (if (<= t_1 2e+180)
            t_2
            (* (fma (* (/ J_m (* U_m U_m)) J_m) -2.0 -1.0) (- U_m)))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double t_2 = (cos((-0.5 * K)) * -2.0) * J_m;
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = -U_m;
	} else if (t_1 <= -1e+212) {
		tmp = t_2;
	} else if (t_1 <= -4e-288) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * (J_m * -2.0);
	} else if (t_1 <= 2e+180) {
		tmp = t_2;
	} else {
		tmp = fma(((J_m / (U_m * U_m)) * J_m), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	t_2 = Float64(Float64(cos(Float64(-0.5 * K)) * -2.0) * J_m)
	tmp = 0.0
	if (t_1 <= -5e+307)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+212)
		tmp = t_2;
	elseif (t_1 <= -4e-288)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(J_m * -2.0));
	elseif (t_1 <= 2e+180)
		tmp = t_2;
	else
		tmp = Float64(fma(Float64(Float64(J_m / Float64(U_m * U_m)) * J_m), -2.0, -1.0) * Float64(-U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * J$95$m), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+307], (-U$95$m), If[LessEqual[t$95$1, -1e+212], t$95$2, If[LessEqual[t$95$1, -4e-288], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+180], t$95$2, N[(N[(N[(N[(J$95$m / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $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))))) < -5e307

   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 45.1

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

   5. Applied rewrites 45.1%

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

   if -5e307 < (*.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))))) < -9.9999999999999991e211 or -4.00000000000000023e-288 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2e180

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 68.0

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

   7. Applied rewrites 68.0%

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

   if -9.9999999999999991e211 < (*.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))))) < -4.00000000000000023e-288

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if 2e180 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 62.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 38.2

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in U around -inf

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

   8. Applied rewrites 25.1%

      \[\leadsto \left(\left(\frac{J}{U \cdot U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(\color{blue}{-2} \cdot J\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 25.1%

       \[\leadsto \left(\left(\frac{-1}{U} \cdot \frac{J}{-U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(-2 \cdot J\right) \]

   11. Taylor expanded in U around -inf

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

   13. Applied rewrites 34.4%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{+212}:\\ \;\;\;\;\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -4 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 2 \cdot 10^{+180}:\\ \;\;\;\;\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U \cdot U} \cdot J, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-158}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m \cdot U\_m} \cdot J\_m, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -5e+307)
      (- U_m)
      (if (<= t_1 -1e-91)
        (* -2.0 J_m)
        (if (<= t_1 2e-158)
          (- U_m)
          (* (fma (* (/ J_m (* U_m U_m)) J_m) -2.0 -1.0) (- U_m))))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = -U_m;
	} else if (t_1 <= -1e-91) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= 2e-158) {
		tmp = -U_m;
	} else {
		tmp = fma(((J_m / (U_m * U_m)) * J_m), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -5e+307)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-91)
		tmp = Float64(-2.0 * J_m);
	elseif (t_1 <= 2e-158)
		tmp = Float64(-U_m);
	else
		tmp = Float64(fma(Float64(Float64(J_m / Float64(U_m * U_m)) * J_m), -2.0, -1.0) * Float64(-U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+307], (-U$95$m), If[LessEqual[t$95$1, -1e-91], N[(-2.0 * J$95$m), $MachinePrecision], If[LessEqual[t$95$1, 2e-158], (-U$95$m), N[(N[(N[(N[(J$95$m / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $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))))) < -5e307 or -1.00000000000000002e-91 < (*.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-158

   1. Initial program 43.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 33.7

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

   5. Applied rewrites 33.7%

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

   if -5e307 < (*.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))))) < -1.00000000000000002e-91

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 75.3

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in K around 0

      \[\leadsto -2 \cdot J \]
   9. Step-by-step derivation

   10. Applied rewrites 52.7%

       \[\leadsto -2 \cdot J \]

   if 2.00000000000000013e-158 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 78.2%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   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. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in U around -inf

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

   8. Applied rewrites 23.6%

      \[\leadsto \left(\left(\frac{J}{U \cdot U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(\color{blue}{-2} \cdot J\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 23.7%

       \[\leadsto \left(\left(\frac{-1}{U} \cdot \frac{J}{-U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(-2 \cdot J\right) \]

   11. Taylor expanded in U around -inf

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

   13. Applied rewrites 29.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 2 \cdot 10^{-158}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U \cdot U} \cdot J, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot J\_m\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-288}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{U\_m}{J\_m} \cdot J\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 -5e+307)
      (- U_m)
      (if (<= t_1 -1e-91)
        (* -2.0 J_m)
        (if (<= t_1 -4e-288) (- U_m) (* (/ U_m J_m) J_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = -U_m;
	} else if (t_1 <= -1e-91) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -4e-288) {
		tmp = -U_m;
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = sqrt((((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0) + 1.0d0)) * (t_0 * (j_m * (-2.0d0)))
    if (t_1 <= (-5d+307)) then
        tmp = -u_m
    else if (t_1 <= (-1d-91)) then
        tmp = (-2.0d0) * j_m
    else if (t_1 <= (-4d-288)) then
        tmp = -u_m
    else
        tmp = (u_m / j_m) * j_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.sqrt((Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -5e+307) {
		tmp = -U_m;
	} else if (t_1 <= -1e-91) {
		tmp = -2.0 * J_m;
	} else if (t_1 <= -4e-288) {
		tmp = -U_m;
	} else {
		tmp = (U_m / J_m) * J_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = math.sqrt((math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0))
	tmp = 0
	if t_1 <= -5e+307:
		tmp = -U_m
	elif t_1 <= -1e-91:
		tmp = -2.0 * J_m
	elif t_1 <= -4e-288:
		tmp = -U_m
	else:
		tmp = (U_m / J_m) * J_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= -5e+307)
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-91)
		tmp = Float64(-2.0 * J_m);
	elseif (t_1 <= -4e-288)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(U_m / J_m) * J_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = sqrt((((U_m / ((2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	tmp = 0.0;
	if (t_1 <= -5e+307)
		tmp = -U_m;
	elseif (t_1 <= -1e-91)
		tmp = -2.0 * J_m;
	elseif (t_1 <= -4e-288)
		tmp = -U_m;
	else
		tmp = (U_m / J_m) * J_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -5e+307], (-U$95$m), If[LessEqual[t$95$1, -1e-91], N[(-2.0 * J$95$m), $MachinePrecision], If[LessEqual[t$95$1, -4e-288], (-U$95$m), N[(N[(U$95$m / J$95$m), $MachinePrecision] * J$95$m), $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))))) < -5e307 or -1.00000000000000002e-91 < (*.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))))) < -4.00000000000000023e-288

   1. Initial program 31.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 36.1

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

   5. Applied rewrites 36.1%

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

   if -5e307 < (*.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))))) < -1.00000000000000002e-91

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 75.3

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in K around 0

      \[\leadsto -2 \cdot J \]
   9. Step-by-step derivation

   10. Applied rewrites 52.7%

       \[\leadsto -2 \cdot J \]

   if -4.00000000000000023e-288 < (*.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 80.0%

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in U around -inf

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

      1. lower-/.f64 25.1

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

   7. Applied rewrites 25.1%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -5 \cdot 10^{+307}:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -1 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -4 \cdot 10^{-288}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\frac{U}{J} \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1}\\ t_2 := t\_1 \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+289}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m \cdot U\_m} \cdot J\_m, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0)))
        (t_2 (* t_1 (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 1e+289)
        (* (* (* (cos (* -0.5 K)) J_m) -2.0) t_1)
        (* (fma (* (/ J_m (* U_m U_m)) J_m) -2.0 -1.0) (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0));
	double t_2 = t_1 * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+289) {
		tmp = ((cos((-0.5 * K)) * J_m) * -2.0) * t_1;
	} else {
		tmp = fma(((J_m / (U_m * U_m)) * J_m), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0))
	t_2 = Float64(t_1 * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+289)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * J_m) * -2.0) * t_1);
	else
		tmp = Float64(fma(Float64(Float64(J_m / Float64(U_m * U_m)) * J_m), -2.0, -1.0) * Float64(-U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+289], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(J$95$m / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $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 45.1

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

   5. Applied rewrites 45.1%

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

   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))))) < 1.0000000000000001e289

   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

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

      2. lift-*.f64 N/A

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

      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. *-commutative N/A

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

      7. lower-*.f64 99.8

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac2 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. div-inv N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.0000000000000001e289 < (*.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 30.5%

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

   3. Taylor expanded in K around 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. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in U around -inf

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

   8. Applied rewrites 34.4%

      \[\leadsto \left(\left(\frac{J}{U \cdot U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(\color{blue}{-2} \cdot J\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 34.4%

       \[\leadsto \left(\left(\frac{-1}{U} \cdot \frac{J}{-U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(-2 \cdot J\right) \]

   11. Taylor expanded in U around -inf

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

   13. Applied rewrites 51.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+289}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U \cdot U} \cdot J, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+289}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{0.5 \cdot U\_m}{\cos \left(0.5 \cdot K\right) \cdot J\_m}\right)}^{2} + 1} \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot -2\right)\right) \cdot J\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m \cdot U\_m} \cdot J\_m, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 1e+289)
        (*
         (*
          (sqrt (+ (pow (/ (* 0.5 U_m) (* (cos (* 0.5 K)) J_m)) 2.0) 1.0))
          (* (cos (* -0.5 K)) -2.0))
         J_m)
        (* (fma (* (/ J_m (* U_m U_m)) J_m) -2.0 -1.0) (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+289) {
		tmp = (sqrt((pow(((0.5 * U_m) / (cos((0.5 * K)) * J_m)), 2.0) + 1.0)) * (cos((-0.5 * K)) * -2.0)) * J_m;
	} else {
		tmp = fma(((J_m / (U_m * U_m)) * J_m), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+289)
		tmp = Float64(Float64(sqrt(Float64((Float64(Float64(0.5 * U_m) / Float64(cos(Float64(0.5 * K)) * J_m)) ^ 2.0) + 1.0)) * Float64(cos(Float64(-0.5 * K)) * -2.0)) * J_m);
	else
		tmp = Float64(fma(Float64(Float64(J_m / Float64(U_m * U_m)) * J_m), -2.0, -1.0) * Float64(-U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+289], N[(N[(N[Sqrt[N[(N[Power[N[(N[(0.5 * U$95$m), $MachinePrecision] / N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision], N[(N[(N[(N[(J$95$m / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $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 45.1

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

   5. Applied rewrites 45.1%

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

   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))))) < 1.0000000000000001e289

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

   if 1.0000000000000001e289 < (*.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 30.5%

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

   3. Taylor expanded in K around 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. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in U around -inf

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

   8. Applied rewrites 34.4%

      \[\leadsto \left(\left(\frac{J}{U \cdot U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(\color{blue}{-2} \cdot J\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 34.4%

       \[\leadsto \left(\left(\frac{-1}{U} \cdot \frac{J}{-U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(-2 \cdot J\right) \]

   11. Taylor expanded in U around -inf

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

   13. Applied rewrites 51.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+289}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{0.5 \cdot U}{\cos \left(0.5 \cdot K\right) \cdot J}\right)}^{2} + 1} \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot -2\right)\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U \cdot U} \cdot J, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ t_2 := \cos \left(0.5 \cdot K\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+237}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(\left(t\_2 \cdot J\_m\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J\_m}{U\_m} \cdot \frac{J\_m}{U\_m}\right) \cdot {t\_2}^{2}, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0))))
        (t_2 (cos (* 0.5 K))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 1e+237)
        (*
         (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0))
         (* (* t_2 J_m) -2.0))
        (*
         (fma (* (* (/ J_m U_m) (/ J_m U_m)) (pow t_2 2.0)) -2.0 -1.0)
         (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double t_2 = cos((0.5 * K));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+237) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * ((t_2 * J_m) * -2.0);
	} else {
		tmp = fma((((J_m / U_m) * (J_m / U_m)) * pow(t_2, 2.0)), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	t_2 = cos(Float64(0.5 * K))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+237)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(Float64(t_2 * J_m) * -2.0));
	else
		tmp = Float64(fma(Float64(Float64(Float64(J_m / U_m) * Float64(J_m / U_m)) * (t_2 ^ 2.0)), -2.0, -1.0) * Float64(-U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+237], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$2 * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(J$95$m / U$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $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 45.1

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

   5. Applied rewrites 45.1%

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

   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))))) < 9.9999999999999994e236

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 86.3%

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

   if 9.9999999999999994e236 < (*.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 55.2%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 37.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+237}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{J}{U} \cdot \frac{J}{U}\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+237}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\_m\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m \cdot U\_m} \cdot J\_m, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 1e+237)
        (*
         (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0))
         (* (* (cos (* 0.5 K)) J_m) -2.0))
        (* (fma (* (/ J_m (* U_m U_m)) J_m) -2.0 -1.0) (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+237) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * ((cos((0.5 * K)) * J_m) * -2.0);
	} else {
		tmp = fma(((J_m / (U_m * U_m)) * J_m), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+237)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(Float64(cos(Float64(0.5 * K)) * J_m) * -2.0));
	else
		tmp = Float64(fma(Float64(Float64(J_m / Float64(U_m * U_m)) * J_m), -2.0, -1.0) * Float64(-U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+237], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(J$95$m / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $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 45.1

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

   5. Applied rewrites 45.1%

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

   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))))) < 9.9999999999999994e236

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 86.3%

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

   if 9.9999999999999994e236 < (*.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 55.2%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   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. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 31.9

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

   5. Applied rewrites 31.9%

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

   6. Taylor expanded in U around -inf

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

   8. Applied rewrites 26.2%

      \[\leadsto \left(\left(\frac{J}{U \cdot U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(\color{blue}{-2} \cdot J\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 26.3%

       \[\leadsto \left(\left(\frac{-1}{U} \cdot \frac{J}{-U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(-2 \cdot J\right) \]

   11. Taylor expanded in U around -inf

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

   13. Applied rewrites 37.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 10^{+237}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U \cdot U} \cdot J, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \sqrt{{\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2} + 1} \cdot \left(t\_0 \cdot \left(J\_m \cdot -2\right)\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U\_m}{J\_m}, \frac{U\_m}{J\_m}, 1\right)} \cdot \left(J\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J\_m}{U\_m \cdot U\_m} \cdot J\_m, -2, -1\right) \cdot \left(-U\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (sqrt (+ (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0) 1.0))
          (* t_0 (* J_m -2.0)))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 2e-158)
        (* (sqrt (fma (/ (* 0.25 U_m) J_m) (/ U_m J_m) 1.0)) (* J_m -2.0))
        (* (fma (* (/ J_m (* U_m U_m)) J_m) -2.0 -1.0) (- U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = sqrt((pow((U_m / ((2.0 * J_m) * t_0)), 2.0) + 1.0)) * (t_0 * (J_m * -2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 2e-158) {
		tmp = sqrt(fma(((0.25 * U_m) / J_m), (U_m / J_m), 1.0)) * (J_m * -2.0);
	} else {
		tmp = fma(((J_m / (U_m * U_m)) * J_m), -2.0, -1.0) * -U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(sqrt(Float64((Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0) + 1.0)) * Float64(t_0 * Float64(J_m * -2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 2e-158)
		tmp = Float64(sqrt(fma(Float64(Float64(0.25 * U_m) / J_m), Float64(U_m / J_m), 1.0)) * Float64(J_m * -2.0));
	else
		tmp = Float64(fma(Float64(Float64(J_m / Float64(U_m * U_m)) * J_m), -2.0, -1.0) * Float64(-U_m));
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 2e-158], N[(N[Sqrt[N[(N[(N[(0.25 * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] * N[(U$95$m / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(J$95$m / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * (-U$95$m)), $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 45.1

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

   5. Applied rewrites 45.1%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 2.00000000000000013e-158

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 2.00000000000000013e-158 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 78.2%

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

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   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. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in U around -inf

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

   8. Applied rewrites 23.6%

      \[\leadsto \left(\left(\frac{J}{U \cdot U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(\color{blue}{-2} \cdot J\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 23.7%

       \[\leadsto \left(\left(\frac{-1}{U} \cdot \frac{J}{-U} + \frac{0.5}{J}\right) \cdot \left(-U\right)\right) \cdot \left(-2 \cdot J\right) \]

   11. Taylor expanded in U around -inf

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

   13. Applied rewrites 29.0%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \leq 2 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25 \cdot U}{J}, \frac{U}{J}, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J}{U \cdot U} \cdot J, -2, -1\right) \cdot \left(-U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.5% accurate, 31.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;J\_m \leq 1.2 \cdot 10^{+54}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= J_m 1.2e+54) (- U_m) (* -2.0 J_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.2e+54) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * J_m;
	}
	return J_s * tmp;
}

Fortran

U_m = abs(u)
J\_m = abs(j)
J\_s = copysign(1.0d0, j)
real(8) function code(j_s, j_m, k, u_m)
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (j_m <= 1.2d+54) then
        tmp = -u_m
    else
        tmp = (-2.0d0) * j_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (J_m <= 1.2e+54) {
		tmp = -U_m;
	} else {
		tmp = -2.0 * J_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if J_m <= 1.2e+54:
		tmp = -U_m
	else:
		tmp = -2.0 * J_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (J_m <= 1.2e+54)
		tmp = Float64(-U_m);
	else
		tmp = Float64(-2.0 * J_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (J_m <= 1.2e+54)
		tmp = -U_m;
	else
		tmp = -2.0 * J_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[J$95$m, 1.2e+54], (-U$95$m), N[(-2.0 * J$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if J < 1.19999999999999999e54

   1. Initial program 70.0%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.6

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

   5. Applied rewrites 27.6%

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

   if 1.19999999999999999e54 < J

   1. Initial program 96.4%

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

   8. Taylor expanded in K around 0

      \[\leadsto -2 \cdot J \]
   9. Step-by-step derivation

   10. Applied rewrites 55.2%

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

Alternative 11: 39.8% accurate, 124.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in 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 24.6

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

5. Applied rewrites 24.6%

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

Reproduce

herbie shell --seed 2024295 
(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)))))