Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.0% → 99.5%

Time: 5.4s

Alternatives: 11

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 73.0% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY)) (- U_m) (if (<= t_1 1e+304) t_1 U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 1e+304) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 1e+304) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 1e+304:
		tmp = t_1
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 1e+304)
		tmp = t_1;
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 1e+304)
		tmp = t_1;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 1e+304], t$95$1, U$95$m]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_3 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{J} \cdot 0.5\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+156}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;\left(\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right) \cdot \left(J \cdot J\right)}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0)))))
        (t_3 (* t_1 (sqrt (+ 1.0 (pow (* (/ U_m J) 0.5) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 -2e+156)
       t_3
       (if (<= t_2 -2e-139)
         (*
          (* (* (cos (* 0.5 K)) J) -2.0)
          (sqrt
           (fma (/ (* U_m U_m) (* (fma (cos K) 0.5 0.5) (* J J))) 0.25 1.0)))
         (if (<= t_2 1e+304) t_3 U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double t_3 = t_1 * sqrt((1.0 + pow(((U_m / J) * 0.5), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -2e+156) {
		tmp = t_3;
	} else if (t_2 <= -2e-139) {
		tmp = ((cos((0.5 * K)) * J) * -2.0) * sqrt(fma(((U_m * U_m) / (fma(cos(K), 0.5, 0.5) * (J * J))), 0.25, 1.0));
	} else if (t_2 <= 1e+304) {
		tmp = t_3;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_3 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(Float64(U_m / J) * 0.5) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -2e+156)
		tmp = t_3;
	elseif (t_2 <= -2e-139)
		tmp = Float64(Float64(Float64(cos(Float64(0.5 * K)) * J) * -2.0) * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(fma(cos(K), 0.5, 0.5) * Float64(J * J))), 0.25, 1.0)));
	elseif (t_2 <= 1e+304)
		tmp = t_3;
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(N[(U$95$m / J), $MachinePrecision] * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -2e+156], t$95$3, If[LessEqual[t$95$2, -2e-139], N[(N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(N[(N[Cos[K], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+304], t$95$3, U$95$m]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2e156 or -2.00000000000000006e-139 < (*.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))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.1

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

   4. Applied rewrites 64.1%

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

   if -2e156 < (*.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))))) < -2.00000000000000006e-139

   1. Initial program 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.7

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

   3. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

   7. Applied rewrites 72.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}}\right)}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \color{blue}{\left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)}\right)}^{2}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}\right)}^{2}} \]

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}}\right)}^{2}} \]

   9. Applied rewrites 72.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}}\right)}^{2}} \]

   10. Taylor expanded in K around inf

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

       1. associate-*r* N/A

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

   12. Applied rewrites 54.7%

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

   13. Taylor expanded in K around inf

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

       1. lower-cos.f64 54.7

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

   15. Applied rewrites 54.7%

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

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{J} \cdot 0.5\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_2 (- INFINITY))
     (- U_m)
     (if (<= t_2 1e+304)
       (* t_1 (sqrt (+ 1.0 (pow (* (/ U_m J) 0.5) 2.0))))
       U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 * sqrt((1.0 + pow(((U_m / J) * 0.5), 2.0)));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = (-2.0 * J) * t_0;
	double t_2 = t_1 * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 * Math.sqrt((1.0 + Math.pow(((U_m / J) * 0.5), 2.0)));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = (-2.0 * J) * t_0
	t_2 = t_1 * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 1e+304:
		tmp = t_1 * math.sqrt((1.0 + math.pow(((U_m / J) * 0.5), 2.0)))
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 1e+304)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(Float64(U_m / J) * 0.5) ^ 2.0))));
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = (-2.0 * J) * t_0;
	t_2 = t_1 * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 1e+304)
		tmp = t_1 * sqrt((1.0 + (((U_m / J) * 0.5) ^ 2.0)));
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 1e+304], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(N[(U$95$m / J), $MachinePrecision] * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.1

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

   4. Applied rewrites 64.1%

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

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(t\_0 \cdot \sqrt{\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J \cdot J}, 0.25, 1\right)}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-139)
       (* (* J -2.0) (* t_0 (sqrt (fma (* U_m (/ U_m (* J J))) 0.25 1.0))))
       (if (<= t_1 1e+304) (* (* J -2.0) (cos (* 0.5 K))) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-139) {
		tmp = (J * -2.0) * (t_0 * sqrt(fma((U_m * (U_m / (J * J))), 0.25, 1.0)));
	} else if (t_1 <= 1e+304) {
		tmp = (J * -2.0) * cos((0.5 * K));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-139)
		tmp = Float64(Float64(J * -2.0) * Float64(t_0 * sqrt(fma(Float64(U_m * Float64(U_m / Float64(J * J))), 0.25, 1.0))));
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(J * -2.0) * cos(Float64(0.5 * K)));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-139], N[(N[(J * -2.0), $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[(J * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000006e-139

   1. Initial program 73.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 50.8

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

   6. Applied rewrites 55.6%

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

   if -2.00000000000000006e-139 < (*.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))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.7

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

   3. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

   7. Applied rewrites 72.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}}\right)}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \color{blue}{\left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)}\right)}^{2}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}\right)}^{2}} \]

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}}\right)}^{2}} \]

   9. Applied rewrites 72.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}}\right)}^{2}} \]

   11. Applied rewrites 84.2%

       \[\leadsto \color{blue}{\left(J \cdot -2\right) \cdot \left(\frac{{\cos \left(\frac{K}{2}\right)}^{3} + 0}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right), \cos \left(\frac{K}{2}\right) - 0, 0\right)} \cdot \cosh \sinh^{-1} \left(\frac{U}{\frac{{\cos \left(\frac{K}{2}\right)}^{3} + 0}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right), \cos \left(\frac{K}{2}\right) - 0, 0\right)} \cdot \left(J + J\right)}\right)\right)} \]

   12. Taylor expanded in J around inf

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 51.1

          \[\leadsto \left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right) \]

   14. Applied rewrites 51.1%

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

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+191}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0)))))
        (t_2 (* (* J -2.0) (cos (* 0.5 K)))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -1e+191)
       t_2
       (if (<= t_1 -2e-139)
         (* (* (sqrt (fma (* U_m (/ U_m (* J J))) 0.25 1.0)) J) -2.0)
         (if (<= t_1 1e+304) t_2 U_m))))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double t_2 = (J * -2.0) * cos((0.5 * K));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e+191) {
		tmp = t_2;
	} else if (t_1 <= -2e-139) {
		tmp = (sqrt(fma((U_m * (U_m / (J * J))), 0.25, 1.0)) * J) * -2.0;
	} else if (t_1 <= 1e+304) {
		tmp = t_2;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_2 = Float64(Float64(J * -2.0) * cos(Float64(0.5 * K)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e+191)
		tmp = t_2;
	elseif (t_1 <= -2e-139)
		tmp = Float64(Float64(sqrt(fma(Float64(U_m * Float64(U_m / Float64(J * J))), 0.25, 1.0)) * J) * -2.0);
	elseif (t_1 <= 1e+304)
		tmp = t_2;
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(J * -2.0), $MachinePrecision] * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e+191], t$95$2, If[LessEqual[t$95$1, -2e-139], N[(N[(N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], t$95$2, U$95$m]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.00000000000000007e191 or -2.00000000000000006e-139 < (*.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))))) < 9.9999999999999994e303

   1. Initial program 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.7

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

   3. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

   7. Applied rewrites 72.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}}\right)}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \color{blue}{\left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)}\right)}^{2}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}\right)}^{2}} \]

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}}\right)}^{2}} \]

   9. Applied rewrites 72.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}}\right)}^{2}} \]

   11. Applied rewrites 84.2%

       \[\leadsto \color{blue}{\left(J \cdot -2\right) \cdot \left(\frac{{\cos \left(\frac{K}{2}\right)}^{3} + 0}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right), \cos \left(\frac{K}{2}\right) - 0, 0\right)} \cdot \cosh \sinh^{-1} \left(\frac{U}{\frac{{\cos \left(\frac{K}{2}\right)}^{3} + 0}{\mathsf{fma}\left(\cos \left(\frac{K}{2}\right), \cos \left(\frac{K}{2}\right) - 0, 0\right)} \cdot \left(J + J\right)}\right)\right)} \]

   12. Taylor expanded in J around inf

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

       1. lift-cos.f64 N/A

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

       2. lift-*.f64 51.1

          \[\leadsto \left(J \cdot -2\right) \cdot \cos \left(0.5 \cdot K\right) \]

   14. Applied rewrites 51.1%

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

   if -1.00000000000000007e191 < (*.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))))) < -2.00000000000000006e-139

   1. Initial program 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.7

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

   3. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

   7. Applied rewrites 72.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}}\right)}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \color{blue}{\left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)}\right)}^{2}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}\right)}^{2}} \]

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}}\right)}^{2}} \]

   9. Applied rewrites 72.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}}\right)}^{2}} \]

   10. Taylor expanded in K around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 36.4%

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

   if 9.9999999999999994e303 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 60.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \frac{U\_m}{J} \cdot 0.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-143}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0)))))
        (t_2 (* (/ U_m J) 0.5)))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -1e-143)
       (* (* (sqrt (fma (* U_m (/ U_m (* J J))) 0.25 1.0)) J) -2.0)
       (if (<= t_1 -1e-262)
         (* (* J -2.0) (* (fma (* K K) -0.125 1.0) (sqrt (fma t_2 t_2 1.0))))
         U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double t_2 = (U_m / J) * 0.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e-143) {
		tmp = (sqrt(fma((U_m * (U_m / (J * J))), 0.25, 1.0)) * J) * -2.0;
	} else if (t_1 <= -1e-262) {
		tmp = (J * -2.0) * (fma((K * K), -0.125, 1.0) * sqrt(fma(t_2, t_2, 1.0)));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	t_2 = Float64(Float64(U_m / J) * 0.5)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-143)
		tmp = Float64(Float64(sqrt(fma(Float64(U_m * Float64(U_m / Float64(J * J))), 0.25, 1.0)) * J) * -2.0);
	elseif (t_1 <= -1e-262)
		tmp = Float64(Float64(J * -2.0) * Float64(fma(Float64(K * K), -0.125, 1.0) * sqrt(fma(t_2, t_2, 1.0))));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(U$95$m / J), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e-143], N[(N[(N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-262], N[(N[(J * -2.0), $MachinePrecision] * N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U$95$m]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -9.9999999999999995e-144

   1. Initial program 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.7

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

   3. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

   7. Applied rewrites 72.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}}\right)}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \color{blue}{\left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)}\right)}^{2}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}\right)}^{2}} \]

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}}\right)}^{2}} \]

   9. Applied rewrites 72.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}}\right)}^{2}} \]

   10. Taylor expanded in K around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 36.4%

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

   if -9.9999999999999995e-144 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 38.0

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

   4. Applied rewrites 38.0%

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

   5. Taylor expanded in K around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 39.8

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

   7. Applied rewrites 39.8%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 38.0

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{J} \cdot 0.5\right)}^{2}} \]

   10. Applied rewrites 38.0%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \mathsf{fma}\left(-0.125, K \cdot K, 1\right)\right) \cdot \sqrt{1 + {\color{blue}{\left(\frac{U}{J} \cdot 0.5\right)}}^{2}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 38.0

          \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \sqrt{1 + {\left(\frac{U}{J} \cdot 0.5\right)}^{2}}\right)} \]

   12. Applied rewrites 38.0%

       \[\leadsto \color{blue}{\left(J \cdot -2\right) \cdot \left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J} \cdot 0.5, \frac{U}{J} \cdot 0.5, 1\right)}\right)} \]

   if -1.00000000000000001e-262 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 59.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(J, -2, \frac{U\_m \cdot U\_m}{J} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-139)
       (* (* (sqrt (fma (* U_m (/ U_m (* J J))) 0.25 1.0)) J) -2.0)
       (if (<= t_1 -1e-262) (fma J -2.0 (* (/ (* U_m U_m) J) -0.25)) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-139) {
		tmp = (sqrt(fma((U_m * (U_m / (J * J))), 0.25, 1.0)) * J) * -2.0;
	} else if (t_1 <= -1e-262) {
		tmp = fma(J, -2.0, (((U_m * U_m) / J) * -0.25));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-139)
		tmp = Float64(Float64(sqrt(fma(Float64(U_m * Float64(U_m / Float64(J * J))), 0.25, 1.0)) * J) * -2.0);
	elseif (t_1 <= -1e-262)
		tmp = fma(J, -2.0, Float64(Float64(Float64(U_m * U_m) / J) * -0.25));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-139], N[(N[(N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-262], N[(J * -2.0 + N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000006e-139

   1. Initial program 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.7

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

   3. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. sin-sum N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-PI.f64 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

   7. Applied rewrites 72.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \mathsf{fma}\left(\sin \left(\frac{K}{2}\right), \cos \left(\frac{\pi}{2}\right), \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)\right)}}\right)}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \color{blue}{\cos \left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \color{blue}{\left(\frac{K}{2}\right)} \cdot \sin \left(\frac{\pi}{2}\right)\right)}\right)}^{2}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \color{blue}{\sin \left(\frac{\pi}{2}\right)}\right)}\right)}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right)}\right)}^{2}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}\right)}\right)}^{2}} \]

      8. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\pi}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}}\right)}^{2}} \]

   9. Applied rewrites 72.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot 0\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot 1\right)}^{3}}{\mathsf{fma}\left(\sin \left(\frac{K}{2}\right) \cdot 0, \sin \left(\frac{K}{2}\right) \cdot 0, \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right) - \left(\sin \left(\frac{K}{2}\right) \cdot 0\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 1\right)\right)}}}\right)}^{2}} \]

   10. Taylor expanded in K around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 36.4%

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

   if -2.00000000000000006e-139 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

         \[\leadsto J \cdot -2 + \frac{-1}{4} \cdot \frac{\color{blue}{{U}^{2}}}{J} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   if -1.00000000000000001e-262 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(J, -2, \frac{U\_m \cdot U\_m}{J} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-139)
       (* (* (sqrt (fma (/ (* U_m U_m) (* J J)) 0.25 1.0)) J) -2.0)
       (if (<= t_1 -1e-262) (fma J -2.0 (* (/ (* U_m U_m) J) -0.25)) U_m)))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e-139) {
		tmp = (sqrt(fma(((U_m * U_m) / (J * J)), 0.25, 1.0)) * J) * -2.0;
	} else if (t_1 <= -1e-262) {
		tmp = fma(J, -2.0, (((U_m * U_m) / J) * -0.25));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e-139)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(U_m * U_m) / Float64(J * J)), 0.25, 1.0)) * J) * -2.0);
	elseif (t_1 <= -1e-262)
		tmp = fma(J, -2.0, Float64(Float64(Float64(U_m * U_m) / J) * -0.25));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-139], N[(N[(N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(J * J), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-262], N[(J * -2.0 + N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], U$95$m]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000006e-139

   1. Initial program 73.0%

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

   2. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 32.2

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

   4. Applied rewrites 32.2%

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

   if -2.00000000000000006e-139 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

         \[\leadsto J \cdot -2 + \frac{-1}{4} \cdot \frac{\color{blue}{{U}^{2}}}{J} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   if -1.00000000000000001e-262 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 54.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(J, -2, \frac{U\_m \cdot U\_m}{J} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -1e-262) (fma J -2.0 (* (/ (* U_m U_m) J) -0.25)) U_m))))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e-262) {
		tmp = fma(J, -2.0, (((U_m * U_m) / J) * -0.25));
	} else {
		tmp = U_m;
	}
	return tmp;
}

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-262)
		tmp = fma(J, -2.0, Float64(Float64(Float64(U_m * U_m) / J) * -0.25));
	else
		tmp = U_m;
	end
	return tmp
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e-262], N[(J * -2.0 + N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / J), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], U$95$m]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. Taylor expanded in U around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

         \[\leadsto J \cdot -2 + \frac{-1}{4} \cdot \frac{\color{blue}{{U}^{2}}}{J} \]

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 27.6

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

   7. Applied rewrites 27.6%

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

   if -1.00000000000000001e-262 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \leq -1 \cdot 10^{-262}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
(FPCore (J K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<=
        (*
         (* (* -2.0 J) t_0)
         (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) t_0)) 2.0))))
        -1e-262)
     (- U_m)
     U_m)))

C

U_m = fabs(U);
double code(double J, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * t_0)), 2.0)))) <= -1e-262) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Fortran

U_m =     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, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (((((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u_m / ((2.0d0 * j) * t_0)) ** 2.0d0)))) <= (-1d-262)) then
        tmp = -u_m
    else
        tmp = u_m
    end if
    code = tmp
end function

Java

U_m = Math.abs(U);
public static double code(double J, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))) <= -1e-262) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return tmp;
}

Python

U_m = math.fabs(U)
def code(J, K, U_m):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * t_0)), 2.0)))) <= -1e-262:
		tmp = -U_m
	else:
		tmp = U_m
	return tmp

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * t_0)) ^ 2.0)))) <= -1e-262)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return tmp
end

MATLAB

U_m = abs(U);
function tmp_2 = code(J, K, U_m)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((((-2.0 * J) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J) * t_0)) ^ 2.0)))) <= -1e-262)
		tmp = -U_m;
	else
		tmp = U_m;
	end
	tmp_2 = tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
code[J_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-262], (-U$95$m), U$95$m]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.00000000000000001e-262

   1. Initial program 73.0%

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

   2. Taylor expanded in J around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.7

         \[\leadsto -U \]

   4. Applied rewrites 26.7%

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

   if -1.00000000000000001e-262 < (*.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 73.0%

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

   2. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   4. Applied rewrites 23.9%

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

   5. Taylor expanded in J around 0

      \[\leadsto U \]
   6. Step-by-step derivation

   7. Applied rewrites 27.5%

      \[\leadsto U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 27.5% accurate, 110.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in U around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower--.f64 N/A

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

4. Applied rewrites 23.9%

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

5. Taylor expanded in J around 0

   \[\leadsto U \]
6. Step-by-step derivation

7. Applied rewrites 27.5%

   \[\leadsto U \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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)))))