Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 99.8%

Time: 9.0s

Alternatives: 10

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    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: 73.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

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

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 (* -0.5 K)))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* (- (fma (/ 2.0 (* U_m U_m)) (* J_m J_m) 1.0)) U_m)
      (if (<= t_2 2e+307)
        (*
         (* (* -2.0 J_m) t_0)
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
        (*
         (fma (* (pow t_0 2.0) (/ (* J_m J_m) (* U_m U_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((-0.5 * K));
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -fma((2.0 / (U_m * U_m)), (J_m * J_m), 1.0) * U_m;
	} else if (t_2 <= 2e+307) {
		tmp = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	} else {
		tmp = fma((pow(t_0, 2.0) * ((J_m * J_m) / (U_m * U_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(-0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-fma(Float64(2.0 / Float64(U_m * U_m)), Float64(J_m * J_m), 1.0)) * U_m);
	elseif (t_2 <= 2e+307)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))));
	else
		tmp = Float64(fma(Float64((t_0 ^ 2.0) * Float64(Float64(J_m * J_m) / Float64(U_m * U_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[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[((-N[(N[(2.0 / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * N[(J$95$m * J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]) * U$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e+307], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -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 5.8%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 41.3%

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

   9. Taylor expanded in J around inf

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

   11. Applied rewrites 41.3%

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

   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))))) < 1.99999999999999997e307

   1. Initial program 99.8%

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

   3. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in K around inf

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

      1. cos-neg-rev N/A

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

      2. lower-cos.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 1.99999999999999997e307 < (*.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 9.5%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-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 48.5%

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

Alternative 2: 59.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(J\_m, \frac{J\_m}{U\_m} \cdot \frac{-2}{U\_m}, -1\right) \cdot U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot U\_m, -0.25, -2 \cdot J\_m\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J\_m}, \frac{U\_m \cdot U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -4e+307)
      (* (fma J_m (* (/ J_m U_m) (/ -2.0 U_m)) -1.0) U_m)
      (if (<= t_1 -1e+165)
        (fma (* (/ U_m J_m) U_m) -0.25 (* -2.0 J_m))
        (if (<= t_1 -2e-168)
          (* (sqrt (fma (/ 0.25 J_m) (/ (* U_m U_m) J_m) 1.0)) (* -2.0 J_m))
          (- 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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -4e+307) {
		tmp = fma(J_m, ((J_m / U_m) * (-2.0 / U_m)), -1.0) * U_m;
	} else if (t_1 <= -1e+165) {
		tmp = fma(((U_m / J_m) * U_m), -0.25, (-2.0 * J_m));
	} else if (t_1 <= -2e-168) {
		tmp = sqrt(fma((0.25 / J_m), ((U_m * U_m) / J_m), 1.0)) * (-2.0 * J_m);
	} else {
		tmp = -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(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -4e+307)
		tmp = Float64(fma(J_m, Float64(Float64(J_m / U_m) * Float64(-2.0 / U_m)), -1.0) * U_m);
	elseif (t_1 <= -1e+165)
		tmp = fma(Float64(Float64(U_m / J_m) * U_m), -0.25, Float64(-2.0 * J_m));
	elseif (t_1 <= -2e-168)
		tmp = Float64(sqrt(fma(Float64(0.25 / J_m), Float64(Float64(U_m * U_m) / J_m), 1.0)) * Float64(-2.0 * J_m));
	else
		tmp = 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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -4e+307], N[(N[(J$95$m * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(-2.0 / U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -1e+165], N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * U$95$m), $MachinePrecision] * -0.25 + N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-168], N[(N[Sqrt[N[(N[(0.25 / J$95$m), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], (-U$95$m)]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999994e307

   1. Initial program 5.8%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 41.3%

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

   10. Applied rewrites 41.3%

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

   if -3.99999999999999994e307 < (*.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))))) < -9.99999999999999899e164

   1. Initial program 99.9%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 9.1

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

   5. Applied rewrites 9.1%

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

   6. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 70.1

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

   8. Applied rewrites 70.1%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 48.6%

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

   13. Applied rewrites 58.8%

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

   if -9.99999999999999899e164 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e-168

   1. Initial program 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

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

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

   1. Initial program 78.1%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 22.8

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

   5. Applied rewrites 22.8%

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

Alternative 3: 90.8% accurate, 0.4× speedup

Math

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

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 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (* (- (fma (/ 2.0 (* U_m U_m)) (* J_m J_m) 1.0)) U_m)
      (if (<= t_2 2e+307)
        (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* 2.0 J_m)) 2.0))))
        (*
         (fma
          (* (pow (cos (* -0.5 K)) 2.0) (/ (* J_m J_m) (* U_m U_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 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -fma((2.0 / (U_m * U_m)), (J_m * J_m), 1.0) * U_m;
	} else if (t_2 <= 2e+307) {
		tmp = t_1 * sqrt((1.0 + pow((U_m / (2.0 * J_m)), 2.0)));
	} else {
		tmp = fma((pow(cos((-0.5 * K)), 2.0) * ((J_m * J_m) / (U_m * U_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(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(-fma(Float64(2.0 / Float64(U_m * U_m)), Float64(J_m * J_m), 1.0)) * U_m);
	elseif (t_2 <= 2e+307)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(2.0 * J_m)) ^ 2.0))));
	else
		tmp = Float64(fma(Float64((cos(Float64(-0.5 * K)) ^ 2.0) * Float64(Float64(J_m * J_m) / Float64(U_m * U_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[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], N[((-N[(N[(2.0 / N[(U$95$m * U$95$m), $MachinePrecision]), $MachinePrecision] * N[(J$95$m * J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]) * U$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e+307], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $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 5.8%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 41.3%

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

   9. Taylor expanded in J around inf

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

   11. Applied rewrites 41.3%

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

   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))))) < 1.99999999999999997e307

   1. Initial program 99.8%

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

   3. Taylor expanded in K around 0

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

      1. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if 1.99999999999999997e307 < (*.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 9.5%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-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 48.5%

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

Alternative 4: 89.1% accurate, 0.4× speedup

Math

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

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -4e+307)
      (* (fma J_m (* (/ J_m U_m) (/ -2.0 U_m)) -1.0) U_m)
      (if (<= t_1 2e+307)
        (*
         (* (* -2.0 J_m) (cos (* 0.5 K)))
         (sqrt (fma (/ 0.25 J_m) (* (/ U_m J_m) U_m) 1.0)))
        (*
         (fma
          (* (pow (cos (* -0.5 K)) 2.0) (/ (* J_m J_m) (* U_m U_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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -4e+307) {
		tmp = fma(J_m, ((J_m / U_m) * (-2.0 / U_m)), -1.0) * U_m;
	} else if (t_1 <= 2e+307) {
		tmp = ((-2.0 * J_m) * cos((0.5 * K))) * sqrt(fma((0.25 / J_m), ((U_m / J_m) * U_m), 1.0));
	} else {
		tmp = fma((pow(cos((-0.5 * K)), 2.0) * ((J_m * J_m) / (U_m * U_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(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -4e+307)
		tmp = Float64(fma(J_m, Float64(Float64(J_m / U_m) * Float64(-2.0 / U_m)), -1.0) * U_m);
	elseif (t_1 <= 2e+307)
		tmp = Float64(Float64(Float64(-2.0 * J_m) * cos(Float64(0.5 * K))) * sqrt(fma(Float64(0.25 / J_m), Float64(Float64(U_m / J_m) * U_m), 1.0)));
	else
		tmp = Float64(fma(Float64((cos(Float64(-0.5 * K)) ^ 2.0) * Float64(Float64(J_m * J_m) / Float64(U_m * U_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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -4e+307], N[(N[(J$95$m * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(-2.0 / U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * U$95$m), $MachinePrecision], If[LessEqual[t$95$1, 2e+307], N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(0.25 / J$95$m), $MachinePrecision] * N[(N[(U$95$m / J$95$m), $MachinePrecision] * U$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(J$95$m * J$95$m), $MachinePrecision] / N[(U$95$m * U$95$m), $MachinePrecision]), $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))))) < -3.99999999999999994e307

   1. Initial program 5.8%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 41.3%

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

   10. Applied rewrites 41.3%

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

   if -3.99999999999999994e307 < (*.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))))) < 1.99999999999999997e307

   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

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in K around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

   10. Applied rewrites 85.4%

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

   if 1.99999999999999997e307 < (*.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 9.5%

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

   3. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. *-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 48.5%

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

Alternative 5: 50.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+307} \lor \neg \left(t\_1 \leq -5 \cdot 10^{-126}\right):\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot U\_m, -0.25, -2 \cdot J\_m\right)\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (or (<= t_1 -4e+307) (not (<= t_1 -5e-126)))
      (- U_m)
      (fma (* (/ U_m J_m) U_m) -0.25 (* -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 t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if ((t_1 <= -4e+307) || !(t_1 <= -5e-126)) {
		tmp = -U_m;
	} else {
		tmp = fma(((U_m / J_m) * U_m), -0.25, (-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)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if ((t_1 <= -4e+307) || !(t_1 <= -5e-126))
		tmp = Float64(-U_m);
	else
		tmp = fma(Float64(Float64(U_m / J_m) * U_m), -0.25, Float64(-2.0 * J_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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[Or[LessEqual[t$95$1, -4e+307], N[Not[LessEqual[t$95$1, -5e-126]], $MachinePrecision]], (-U$95$m), N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * U$95$m), $MachinePrecision] * -0.25 + N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999994e307 or -5.00000000000000006e-126 < (*.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 60.4%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 27.5

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

   5. Applied rewrites 27.5%

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

   if -3.99999999999999994e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000006e-126

   1. Initial program 99.9%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 14.4

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

   5. Applied rewrites 14.4%

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

   6. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 76.5

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 44.2%

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

   13. Applied rewrites 47.4%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -4 \cdot 10^{+307} \lor \neg \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -5 \cdot 10^{-126}\right):\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U}{J} \cdot U, -0.25, -2 \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(J\_m, \frac{J\_m}{U\_m} \cdot \frac{-2}{U\_m}, -1\right) \cdot U\_m\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U\_m}{J\_m} \cdot U\_m, -0.25, -2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -4e+307)
      (* (fma J_m (* (/ J_m U_m) (/ -2.0 U_m)) -1.0) U_m)
      (if (<= t_1 -5e-126)
        (fma (* (/ U_m J_m) U_m) -0.25 (* -2.0 J_m))
        (- 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 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -4e+307) {
		tmp = fma(J_m, ((J_m / U_m) * (-2.0 / U_m)), -1.0) * U_m;
	} else if (t_1 <= -5e-126) {
		tmp = fma(((U_m / J_m) * U_m), -0.25, (-2.0 * J_m));
	} else {
		tmp = -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(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -4e+307)
		tmp = Float64(fma(J_m, Float64(Float64(J_m / U_m) * Float64(-2.0 / U_m)), -1.0) * U_m);
	elseif (t_1 <= -5e-126)
		tmp = fma(Float64(Float64(U_m / J_m) * U_m), -0.25, Float64(-2.0 * J_m));
	else
		tmp = 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[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -4e+307], N[(N[(J$95$m * N[(N[(J$95$m / U$95$m), $MachinePrecision] * N[(-2.0 / U$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * U$95$m), $MachinePrecision], If[LessEqual[t$95$1, -5e-126], N[(N[(N[(U$95$m / J$95$m), $MachinePrecision] * U$95$m), $MachinePrecision] * -0.25 + N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], (-U$95$m)]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999994e307

   1. Initial program 5.8%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 41.3%

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

   10. Applied rewrites 41.3%

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

   if -3.99999999999999994e307 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -5.00000000000000006e-126

   1. Initial program 99.9%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 14.4

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

   5. Applied rewrites 14.4%

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

   6. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 76.5

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in K around 0

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

   11. Applied rewrites 44.2%

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

   13. Applied rewrites 47.4%

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

   if -5.00000000000000006e-126 < (*.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 78.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 23.0

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

   5. Applied rewrites 23.0%

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

Alternative 7: 82.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999994e307

   1. Initial program 5.8%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 41.3%

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

   10. Applied rewrites 41.3%

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

   if -3.99999999999999994e307 < (*.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 87.6%

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

   3. Taylor expanded in K around 0

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

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in K around inf

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   10. Applied rewrites 74.8%

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

Alternative 8: 72.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999994e307

   1. Initial program 5.8%

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

   3. Taylor expanded in U around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in K around 0

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

   8. Applied rewrites 41.3%

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

   10. Applied rewrites 41.3%

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

   if -3.99999999999999994e307 < (*.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 87.6%

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

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in J around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

Alternative 9: 47.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}} \leq -2 \cdot 10^{+304}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(K \cdot K\right) \cdot 0.25 - 2\right) \cdot J\_m\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    J_s
    (if (<=
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))
         -2e+304)
      (- U_m)
      (* (- (* (* K K) 0.25) 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 t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -2e+304) {
		tmp = -U_m;
	} else {
		tmp = (((K * K) * 0.25) - 2.0) * J_m;
	}
	return J_s * tmp;
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    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) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((((-2.0d0) * j_m) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j_m) * t_0)) ** 2.0d0)))) <= (-2d+304)) then
        tmp = -u_m
    else
        tmp = (((k * k) * 0.25d0) - 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 t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -2e+304) {
		tmp = -U_m;
	} else {
		tmp = (((K * K) * 0.25) - 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):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))) <= -2e+304:
		tmp = -U_m
	else:
		tmp = (((K * K) * 0.25) - 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)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0)))) <= -2e+304)
		tmp = Float64(-U_m);
	else
		tmp = Float64(Float64(Float64(Float64(K * K) * 0.25) - 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)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)))) <= -2e+304)
		tmp = -U_m;
	else
		tmp = (((K * K) * 0.25) - 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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(J$95$s * If[LessEqual[N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2e+304], (-U$95$m), N[(N[(N[(N[(K * K), $MachinePrecision] * 0.25), $MachinePrecision] - 2.0), $MachinePrecision] * J$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.9999999999999999e304

   1. Initial program 5.8%

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

   3. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 41.0

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

   5. Applied rewrites 41.0%

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

   if -1.9999999999999999e304 < (*.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 87.6%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   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 19.4

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

   5. Applied rewrites 19.4%

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

   6. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 62.7

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

   8. Applied rewrites 62.7%

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

   9. Taylor expanded in K around 0

      \[\leadsto -2 \cdot J + \color{blue}{\left(\frac{-1}{4} \cdot \frac{{U}^{2}}{J} + {K}^{2} \cdot \left(\frac{-1}{32} \cdot \frac{{U}^{2}}{J} + \frac{1}{4} \cdot J\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.1%

       \[\leadsto \mathsf{fma}\left(-2, \color{blue}{J}, \mathsf{fma}\left(\mathsf{fma}\left(-0.03125, \frac{U \cdot U}{J}, 0.25 \cdot J\right), K \cdot K, \frac{U \cdot U}{J} \cdot -0.25\right)\right) \]

   12. Taylor expanded in J around inf

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

   14. Applied rewrites 31.9%

       \[\leadsto \left(\left(K \cdot K\right) \cdot 0.25 - 2\right) \cdot J \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 39.0% accurate, 124.3× speedup

Math

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

FPCore

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

C

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

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    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 74.1%

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

3. Taylor expanded in J around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 22.9

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

5. Applied rewrites 22.9%

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

Reproduce

herbie shell --seed 2025009 
(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)))))