Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.0% → 99.5%

Time: 4.7s

Alternatives: 10

Speedup: 0.3×

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 5.9%

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

   2. 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 99.8

         \[\leadsto -U \]

   4. Applied rewrites 99.8%

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

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

   1. Initial program 99.8%

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

   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 8.0%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 97.7%

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

Alternative 2: 83.3% accurate, 0.2× speedup

Math

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

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 (* (* (cos (* 0.5 K)) J) -2.0)))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e+156)
       (*
        (* (* (cos (* K 0.5)) J) -2.0)
        (sqrt (fma (* U_m (/ U_m (* J J))) 0.25 1.0)))
       (if (<= t_1 -2e-139)
         (*
          t_2
          (sqrt
           (fma
            (/ (* U_m U_m) (* (fma (cos (* 2.0 (* 0.5 K))) 0.5 0.5) (* J J)))
            0.25
            1.0)))
         (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 = (cos((0.5 * K)) * J) * -2.0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e+156) {
		tmp = ((cos((K * 0.5)) * J) * -2.0) * sqrt(fma((U_m * (U_m / (J * J))), 0.25, 1.0));
	} else if (t_1 <= -2e-139) {
		tmp = t_2 * sqrt(fma(((U_m * U_m) / (fma(cos((2.0 * (0.5 * K))), 0.5, 0.5) * (J * J))), 0.25, 1.0));
	} 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(cos(Float64(0.5 * K)) * J) * -2.0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e+156)
		tmp = Float64(Float64(Float64(cos(Float64(K * 0.5)) * J) * -2.0) * sqrt(fma(Float64(U_m * Float64(U_m / Float64(J * J))), 0.25, 1.0)));
	elseif (t_1 <= -2e-139)
		tmp = Float64(t_2 * sqrt(fma(Float64(Float64(U_m * U_m) / Float64(fma(cos(Float64(2.0 * Float64(0.5 * K))), 0.5, 0.5) * Float64(J * J))), 0.25, 1.0)));
	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[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e+156], N[(N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-139], N[(t$95$2 * N[Sqrt[N[(N[(N[(U$95$m * U$95$m), $MachinePrecision] / N[(N[(N[Cos[N[(2.0 * N[(0.5 * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], t$95$2, U$95$m]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 5.9%

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

   2. 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 99.8

         \[\leadsto -U \]

   4. Applied rewrites 99.8%

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

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

   1. Initial program 99.8%

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

   2. 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 67.5

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

   4. Applied rewrites 67.5%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)}} \]
   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. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 67.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 67.5

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

   6. Applied rewrites 83.4%

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

   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 99.8%

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. unpow1 N/A

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

      7. metadata-eval N/A

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

      8. pow-neg N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lift-cos.f64 N/A

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

      16. mult-flip N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.7

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

   3. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. mult-flip N/A

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. unpow1 N/A

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

      7. metadata-eval N/A

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

      8. pow-neg N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lift-cos.f64 N/A

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

      16. mult-flip N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in K around inf

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

   10. Applied rewrites 92.7%

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

   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 99.8%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-cos.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.5

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

   4. Applied rewrites 67.5%

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

   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 8.0%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 97.7%

      \[\leadsto \color{blue}{U} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-58}:\\ \;\;\;\;\left(\left(\cos \left(K \cdot 0.5\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(U\_m \cdot \frac{U\_m}{J \cdot J}, 0.25, 1\right)}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-180}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot 1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot 1}\right)}^{2}}\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -2e-58)
       (*
        (* (* (cos (* K 0.5)) J) -2.0)
        (sqrt (fma (* U_m (/ U_m (* J J))) 0.25 1.0)))
       (if (<= t_1 -4e-180)
         (*
          (* (* -2.0 J) 1.0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) 1.0)) 2.0))))
         (if (<= t_1 1e+304) (* (* (cos (* 0.5 K)) J) -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) * 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-58) {
		tmp = ((cos((K * 0.5)) * J) * -2.0) * sqrt(fma((U_m * (U_m / (J * J))), 0.25, 1.0));
	} else if (t_1 <= -4e-180) {
		tmp = ((-2.0 * J) * 1.0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * 1.0)), 2.0)));
	} else if (t_1 <= 1e+304) {
		tmp = (cos((0.5 * K)) * J) * -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(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-58)
		tmp = Float64(Float64(Float64(cos(Float64(K * 0.5)) * J) * -2.0) * sqrt(fma(Float64(U_m * Float64(U_m / Float64(J * J))), 0.25, 1.0)));
	elseif (t_1 <= -4e-180)
		tmp = Float64(Float64(Float64(-2.0 * J) * 1.0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * 1.0)) ^ 2.0))));
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(cos(Float64(0.5 * K)) * J) * -2.0);
	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]}, If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e-58], N[(N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(N[(U$95$m * N[(U$95$m / N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-180], N[(N[(N[(-2.0 * J), $MachinePrecision] * 1.0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision], U$95$m]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 5.9%

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

   2. 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 99.8

         \[\leadsto -U \]

   4. Applied rewrites 99.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e-58

   1. Initial program 99.8%

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

   2. 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 77.4

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

   4. Applied rewrites 77.4%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot U}{J \cdot J}, 0.25, 1\right)}} \]
   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. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 77.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 77.4

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

   6. Applied rewrites 85.3%

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

   if -2.0000000000000001e-58 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000001e-180

   1. Initial program 99.8%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 62.9%

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

   if -4.0000000000000001e-180 < (*.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 99.8%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-cos.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.2

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

   4. Applied rewrites 68.2%

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

   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 8.0%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 97.7%

      \[\leadsto \color{blue}{U} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-180}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot 1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot 1}\right)}^{2}}\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;\left(\cos \left(0.5 \cdot K\right) \cdot J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \]

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))))))
   (if (<= t_1 (- INFINITY))
     (- U_m)
     (if (<= t_1 -4e-180)
       (*
        (* (* -2.0 J) 1.0)
        (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J) 1.0)) 2.0))))
       (if (<= t_1 1e+304) (* (* (cos (* 0.5 K)) J) -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) * 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 <= -4e-180) {
		tmp = ((-2.0 * J) * 1.0) * sqrt((1.0 + pow((U_m / ((2.0 * J) * 1.0)), 2.0)));
	} else if (t_1 <= 1e+304) {
		tmp = (cos((0.5 * K)) * J) * -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) * 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 <= -4e-180) {
		tmp = ((-2.0 * J) * 1.0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J) * 1.0)), 2.0)));
	} else if (t_1 <= 1e+304) {
		tmp = (Math.cos((0.5 * K)) * J) * -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) * 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 <= -4e-180:
		tmp = ((-2.0 * J) * 1.0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J) * 1.0)), 2.0)))
	elif t_1 <= 1e+304:
		tmp = (math.cos((0.5 * K)) * J) * -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(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 <= -4e-180)
		tmp = Float64(Float64(Float64(-2.0 * J) * 1.0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * 1.0)) ^ 2.0))));
	elseif (t_1 <= 1e+304)
		tmp = Float64(Float64(cos(Float64(0.5 * K)) * J) * -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) * 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 <= -4e-180)
		tmp = ((-2.0 * J) * 1.0) * sqrt((1.0 + ((U_m / ((2.0 * J) * 1.0)) ^ 2.0)));
	elseif (t_1 <= 1e+304)
		tmp = (cos((0.5 * K)) * J) * -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[(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, -4e-180], N[(N[(N[(-2.0 * J), $MachinePrecision] * 1.0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $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 5.9%

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

   2. 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 99.8

         \[\leadsto -U \]

   4. Applied rewrites 99.8%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000001e-180

   1. Initial program 99.8%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 60.9%

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

   if -4.0000000000000001e-180 < (*.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 99.8%

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

   2. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-cos.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.2

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

   4. Applied rewrites 68.2%

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

   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 8.0%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 97.7%

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

Alternative 5: 63.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(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 = Float64(Float64(Float64(-2.0 * J) * 1.0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J) * 1.0)) ^ 2.0))));
	else
		tmp = Float64(Float64(-U_m) * fma(Float64(Float64(J / U_m) * Float64(J / U_m)), -2.0, -1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.9%

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

   2. 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 99.8

         \[\leadsto -U \]

   4. Applied rewrites 99.8%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in K around 0

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in K around 0

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

   7. Applied rewrites 60.8%

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

   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 73.1%

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

   2. 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 31.6

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-flip N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 47.0

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

   7. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 54.1

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

   9. Applied rewrites 54.1%

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

Alternative 6: 57.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-139}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J \cdot J}, 0.25, 1\right)} \cdot J\right) \cdot -2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\mathsf{fma}\left(\frac{U\_m \cdot U\_m}{J}, 0.125, J\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-U\_m\right) \cdot \mathsf{fma}\left(\frac{J}{U\_m} \cdot \frac{J}{U\_m}, -2, -1\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(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 = Float64(fma(Float64(Float64(U_m * U_m) / J), 0.125, J) * -2.0);
	else
		tmp = Float64(Float64(-U_m) * fma(Float64(Float64(J / U_m) * Float64(J / U_m)), -2.0, -1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 5.9%

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

   2. 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 99.8

         \[\leadsto -U \]

   4. Applied rewrites 99.8%

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

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

   1. Initial program 99.8%

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

   2. 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 48.2

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

   4. Applied rewrites 48.2%

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

   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 99.8%

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

   2. 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 7.5

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

   4. Applied rewrites 7.5%

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

   5. Taylor expanded in U around 0

      \[\leadsto \left(J + \frac{1}{8} \cdot \frac{{U}^{2}}{J}\right) \cdot -2 \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{U \cdot U}{J}, \frac{1}{8}, J\right) \cdot -2 \]

      6. lift-*.f64 26.0

         \[\leadsto \mathsf{fma}\left(\frac{U \cdot U}{J}, 0.125, J\right) \cdot -2 \]

   7. Applied rewrites 26.0%

      \[\leadsto \mathsf{fma}\left(\frac{U \cdot U}{J}, 0.125, J\right) \cdot -2 \]

   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 73.1%

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

   2. 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 31.6

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-flip N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 47.0

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

   7. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 54.1

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

   9. Applied rewrites 54.1%

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

Alternative 7: 55.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

U_m = abs(U)
function code(J, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(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 = Float64(J * -2.0);
	else
		tmp = Float64(Float64(-U_m) * fma(Float64(Float64(J / U_m) * Float64(J / U_m)), -2.0, -1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.9%

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

   2. 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 99.8

         \[\leadsto -U \]

   4. Applied rewrites 99.8%

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

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

   1. Initial program 99.8%

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

   2. 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 44.3

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

   4. Applied rewrites 44.3%

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

   5. Taylor expanded in J around inf

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

   7. Applied rewrites 39.6%

      \[\leadsto J \cdot -2 \]

   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 73.1%

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

   2. 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 31.6

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in U around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-flip N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 47.0

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

   7. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 54.1

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

   9. Applied rewrites 54.1%

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

Alternative 8: 55.1% accurate, 0.5× speedup

Math

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

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))))))
   (if (<= t_1 (- INFINITY)) (- U_m) (if (<= t_1 -1e-262) (* J -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) * 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 = J * -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) * 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-262) {
		tmp = J * -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) * 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-262:
		tmp = J * -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(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 = Float64(J * -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) * 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-262)
		tmp = J * -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[(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), $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 5.9%

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

   2. 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 99.8

         \[\leadsto -U \]

   4. Applied rewrites 99.8%

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

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

   1. Initial program 99.8%

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

   2. 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 44.3

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

   4. Applied rewrites 44.3%

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

   5. Taylor expanded in J around inf

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

   7. Applied rewrites 39.6%

      \[\leadsto J \cdot -2 \]

   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 73.1%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 53.3%

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

Alternative 9: 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 72.8%

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

   2. 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 52.1

         \[\leadsto -U \]

   4. Applied rewrites 52.1%

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

   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 73.1%

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

   2. Taylor expanded in U around -inf

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

   4. Applied rewrites 53.3%

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

Alternative 10: 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

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

4. Applied rewrites 27.5%

   \[\leadsto \color{blue}{U} \]
5. 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)))))