Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.1% → 82.3%

Time: 28.5s

Alternatives: 7

Speedup: 99.6×

• 33.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 25.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 82.3% accurate, 52.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 2.8 \cdot 10^{-275}:\\ \;\;\;\;\frac{a\_m}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{-4 \cdot b}} \cdot \left(a\_m \cdot b\right)\\ \mathbf{elif}\;a\_m \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{a\_m}{\frac{y-scale}{-4 \cdot \frac{b}{\frac{x-scale}{b}}}}}{\frac{y-scale}{\frac{a\_m}{x-scale}}}\\ \mathbf{elif}\;a\_m \leq 2.9 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot \left(b \cdot \left(a\_m \cdot a\_m\right)\right)}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{x-scale} \cdot \left(b \cdot \frac{a\_m \cdot \frac{\frac{-4}{y-scale}}{y-scale}}{\frac{x-scale}{b}}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 2.8e-275)
   (*
    (/ a_m (/ (* (* y-scale x-scale) (* y-scale x-scale)) (* -4.0 b)))
    (* a_m b))
   (if (<= a_m 1.7e-127)
     (/
      (/ a_m (/ y-scale (* -4.0 (/ b (/ x-scale b)))))
      (/ y-scale (/ a_m x-scale)))
     (if (<= a_m 2.9e+153)
       (*
        (/ b (* y-scale x-scale))
        (/ (* -4.0 (* b (* a_m a_m))) (* y-scale x-scale)))
       (*
        (/ a_m x-scale)
        (* b (/ (* a_m (/ (/ -4.0 y-scale) y-scale)) (/ x-scale b))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 2.8e-275) {
		tmp = (a_m / (((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)) / (-4.0 * b))) * (a_m * b);
	} else if (a_m <= 1.7e-127) {
		tmp = (a_m / (y_45_scale / (-4.0 * (b / (x_45_scale / b))))) / (y_45_scale / (a_m / x_45_scale));
	} else if (a_m <= 2.9e+153) {
		tmp = (b / (y_45_scale * x_45_scale)) * ((-4.0 * (b * (a_m * a_m))) / (y_45_scale * x_45_scale));
	} else {
		tmp = (a_m / x_45_scale) * (b * ((a_m * ((-4.0 / y_45_scale) / y_45_scale)) / (x_45_scale / b)));
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (a_m <= 2.8d-275) then
        tmp = (a_m / (((y_45scale * x_45scale) * (y_45scale * x_45scale)) / ((-4.0d0) * b))) * (a_m * b)
    else if (a_m <= 1.7d-127) then
        tmp = (a_m / (y_45scale / ((-4.0d0) * (b / (x_45scale / b))))) / (y_45scale / (a_m / x_45scale))
    else if (a_m <= 2.9d+153) then
        tmp = (b / (y_45scale * x_45scale)) * (((-4.0d0) * (b * (a_m * a_m))) / (y_45scale * x_45scale))
    else
        tmp = (a_m / x_45scale) * (b * ((a_m * (((-4.0d0) / y_45scale) / y_45scale)) / (x_45scale / b)))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 2.8e-275) {
		tmp = (a_m / (((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)) / (-4.0 * b))) * (a_m * b);
	} else if (a_m <= 1.7e-127) {
		tmp = (a_m / (y_45_scale / (-4.0 * (b / (x_45_scale / b))))) / (y_45_scale / (a_m / x_45_scale));
	} else if (a_m <= 2.9e+153) {
		tmp = (b / (y_45_scale * x_45_scale)) * ((-4.0 * (b * (a_m * a_m))) / (y_45_scale * x_45_scale));
	} else {
		tmp = (a_m / x_45_scale) * (b * ((a_m * ((-4.0 / y_45_scale) / y_45_scale)) / (x_45_scale / b)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 2.8e-275:
		tmp = (a_m / (((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)) / (-4.0 * b))) * (a_m * b)
	elif a_m <= 1.7e-127:
		tmp = (a_m / (y_45_scale / (-4.0 * (b / (x_45_scale / b))))) / (y_45_scale / (a_m / x_45_scale))
	elif a_m <= 2.9e+153:
		tmp = (b / (y_45_scale * x_45_scale)) * ((-4.0 * (b * (a_m * a_m))) / (y_45_scale * x_45_scale))
	else:
		tmp = (a_m / x_45_scale) * (b * ((a_m * ((-4.0 / y_45_scale) / y_45_scale)) / (x_45_scale / b)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 2.8e-275)
		tmp = Float64(Float64(a_m / Float64(Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)) / Float64(-4.0 * b))) * Float64(a_m * b));
	elseif (a_m <= 1.7e-127)
		tmp = Float64(Float64(a_m / Float64(y_45_scale / Float64(-4.0 * Float64(b / Float64(x_45_scale / b))))) / Float64(y_45_scale / Float64(a_m / x_45_scale)));
	elseif (a_m <= 2.9e+153)
		tmp = Float64(Float64(b / Float64(y_45_scale * x_45_scale)) * Float64(Float64(-4.0 * Float64(b * Float64(a_m * a_m))) / Float64(y_45_scale * x_45_scale)));
	else
		tmp = Float64(Float64(a_m / x_45_scale) * Float64(b * Float64(Float64(a_m * Float64(Float64(-4.0 / y_45_scale) / y_45_scale)) / Float64(x_45_scale / b))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 2.8e-275)
		tmp = (a_m / (((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)) / (-4.0 * b))) * (a_m * b);
	elseif (a_m <= 1.7e-127)
		tmp = (a_m / (y_45_scale / (-4.0 * (b / (x_45_scale / b))))) / (y_45_scale / (a_m / x_45_scale));
	elseif (a_m <= 2.9e+153)
		tmp = (b / (y_45_scale * x_45_scale)) * ((-4.0 * (b * (a_m * a_m))) / (y_45_scale * x_45_scale));
	else
		tmp = (a_m / x_45_scale) * (b * ((a_m * ((-4.0 / y_45_scale) / y_45_scale)) / (x_45_scale / b)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 2.8e-275], N[(N[(a$95$m / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 1.7e-127], N[(N[(a$95$m / N[(y$45$scale / N[(-4.0 * N[(b / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale / N[(a$95$m / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 2.9e+153], N[(N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * N[(b * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / x$45$scale), $MachinePrecision] * N[(b * N[(N[(a$95$m * N[(N[(-4.0 / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < 2.79999999999999994e-275

   1. Initial program 26.6%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

      17. *-lowering-*.f64 45.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

   5. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 55.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

   7. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale}\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{1}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \left(\frac{x-scale}{b}\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      12. /-lowering-/.f64 71.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \mathsf{/.f64}\left(x-scale, b\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. Applied egg-rr 71.8%

      \[\leadsto \left(\color{blue}{\frac{\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}}{y-scale}} \cdot a\right) \cdot \frac{a}{x-scale} \]

   11. Applied egg-rr 74.7%

       \[\leadsto \color{blue}{\frac{a}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{-4 \cdot b}} \cdot \left(b \cdot a\right)} \]

   if 2.79999999999999994e-275 < a < 1.6999999999999999e-127

   1. Initial program 43.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

      17. *-lowering-*.f64 48.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

   5. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 65.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

   7. Applied egg-rr 65.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale}\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{1}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \left(\frac{x-scale}{b}\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      12. /-lowering-/.f64 90.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \mathsf{/.f64}\left(x-scale, b\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. Applied egg-rr 90.7%

      \[\leadsto \left(\color{blue}{\frac{\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}}{y-scale}} \cdot a\right) \cdot \frac{a}{x-scale} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \left(\frac{\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}}{y-scale} \cdot a\right) \cdot \frac{1}{\color{blue}{\frac{x-scale}{a}}} \]

       2. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}}{y-scale} \cdot a}{\color{blue}{\frac{x-scale}{a}}} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\frac{\left(\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}\right) \cdot a}{y-scale}}{\frac{\color{blue}{x-scale}}{a}} \]

       4. associate-/l/ N/A

          \[\leadsto \frac{\left(\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}\right) \cdot a}{\color{blue}{\frac{x-scale}{a} \cdot y-scale}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}\right) \cdot a\right), \color{blue}{\left(\frac{x-scale}{a} \cdot y-scale\right)}\right) \]

   11. Applied egg-rr 90.8%

       \[\leadsto \color{blue}{\frac{\frac{a}{\frac{y-scale}{-4 \cdot \frac{b}{\frac{x-scale}{b}}}}}{\frac{y-scale}{\frac{a}{x-scale}}}} \]

   if 1.6999999999999999e-127 < a < 2.90000000000000002e153

   1. Initial program 35.4%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

      17. *-lowering-*.f64 66.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 70.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

   7. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot \color{blue}{\left(a \cdot \frac{a}{x-scale}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot \frac{a \cdot a}{\color{blue}{x-scale}} \]

      3. div-inv N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right) \cdot \frac{\color{blue}{a \cdot a}}{x-scale} \]

      4. clear-num N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \frac{a \cdot \color{blue}{a}}{x-scale} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \color{blue}{\left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]

      6. frac-times N/A

         \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot x-scale}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot \left(b \cdot \left(a \cdot a\right)\right)}{\color{blue}{x-scale} \cdot x-scale} \]

      8. times-frac N/A

         \[\leadsto \frac{-4 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right) \cdot -4}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot \left(x-scale \cdot x-scale\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{b \cdot \left(\left(b \cdot \left(a \cdot a\right)\right) \cdot -4\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot \left(x-scale \cdot x-scale\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{b \cdot \left(\left(b \cdot \left(a \cdot a\right)\right) \cdot -4\right)}{\left(x-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}} \]

      12. swap-sqr N/A

          \[\leadsto \frac{b \cdot \left(\left(b \cdot \left(a \cdot a\right)\right) \cdot -4\right)}{\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}} \]

      13. times-frac N/A

          \[\leadsto \frac{b}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot -4}{x-scale \cdot y-scale}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{b}{x-scale \cdot y-scale}\right), \color{blue}{\left(\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot -4}{x-scale \cdot y-scale}\right)}\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \left(x-scale \cdot y-scale\right)\right), \left(\frac{\color{blue}{\left(b \cdot \left(a \cdot a\right)\right) \cdot -4}}{x-scale \cdot y-scale}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \left(y-scale \cdot x-scale\right)\right), \left(\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{-4}}{x-scale \cdot y-scale}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(y-scale, x-scale\right)\right), \left(\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{-4}}{x-scale \cdot y-scale}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(y-scale, x-scale\right)\right), \mathsf{/.f64}\left(\left(\left(b \cdot \left(a \cdot a\right)\right) \cdot -4\right), \color{blue}{\left(x-scale \cdot y-scale\right)}\right)\right) \]

   9. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\frac{b}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot \left(b \cdot \left(a \cdot a\right)\right)}{y-scale \cdot x-scale}} \]

   if 2.90000000000000002e153 < a

   1. Initial program 0.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

      17. *-lowering-*.f64 47.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

   5. Simplified 47.9%

      \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 69.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

   7. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale} \cdot a}{\frac{x-scale}{b \cdot b}}\right), \mathsf{/.f64}\left(\color{blue}{a}, x-scale\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale} \cdot a}{\frac{\frac{x-scale}{b}}{b}}\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale} \cdot a}{\frac{x-scale}{b}} \cdot b\right), \mathsf{/.f64}\left(\color{blue}{a}, x-scale\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale} \cdot a}{\frac{x-scale}{b}}\right), b\right), \mathsf{/.f64}\left(\color{blue}{a}, x-scale\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale}\right), a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale}\right), y-scale\right), a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), y-scale\right), a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      10. /-lowering-/.f64 86.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), y-scale\right), a\right), \mathsf{/.f64}\left(x-scale, b\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{\left(\frac{\frac{\frac{-4}{y-scale}}{y-scale} \cdot a}{\frac{x-scale}{b}} \cdot b\right)} \cdot \frac{a}{x-scale} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-275}:\\ \;\;\;\;\frac{a}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{-4 \cdot b}} \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{a}{\frac{y-scale}{-4 \cdot \frac{b}{\frac{x-scale}{b}}}}}{\frac{y-scale}{\frac{a}{x-scale}}}\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot \left(b \cdot \left(a \cdot a\right)\right)}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{x-scale} \cdot \left(b \cdot \frac{a \cdot \frac{\frac{-4}{y-scale}}{y-scale}}{\frac{x-scale}{b}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 62.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;\frac{a\_m}{x-scale} \cdot \left(a\_m \cdot \frac{\frac{\frac{b}{\frac{y-scale}{-4}}}{x-scale}}{\frac{y-scale}{b}}\right)\\ \mathbf{elif}\;a\_m \leq 2.9 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot \left(b \cdot \left(a\_m \cdot a\_m\right)\right)}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{x-scale} \cdot \left(b \cdot \frac{a\_m \cdot \frac{\frac{-4}{y-scale}}{y-scale}}{\frac{x-scale}{b}}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 1.7e-127)
   (*
    (/ a_m x-scale)
    (* a_m (/ (/ (/ b (/ y-scale -4.0)) x-scale) (/ y-scale b))))
   (if (<= a_m 2.9e+153)
     (*
      (/ b (* y-scale x-scale))
      (/ (* -4.0 (* b (* a_m a_m))) (* y-scale x-scale)))
     (*
      (/ a_m x-scale)
      (* b (/ (* a_m (/ (/ -4.0 y-scale) y-scale)) (/ x-scale b)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 1.7e-127) {
		tmp = (a_m / x_45_scale) * (a_m * (((b / (y_45_scale / -4.0)) / x_45_scale) / (y_45_scale / b)));
	} else if (a_m <= 2.9e+153) {
		tmp = (b / (y_45_scale * x_45_scale)) * ((-4.0 * (b * (a_m * a_m))) / (y_45_scale * x_45_scale));
	} else {
		tmp = (a_m / x_45_scale) * (b * ((a_m * ((-4.0 / y_45_scale) / y_45_scale)) / (x_45_scale / b)));
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (a_m <= 1.7d-127) then
        tmp = (a_m / x_45scale) * (a_m * (((b / (y_45scale / (-4.0d0))) / x_45scale) / (y_45scale / b)))
    else if (a_m <= 2.9d+153) then
        tmp = (b / (y_45scale * x_45scale)) * (((-4.0d0) * (b * (a_m * a_m))) / (y_45scale * x_45scale))
    else
        tmp = (a_m / x_45scale) * (b * ((a_m * (((-4.0d0) / y_45scale) / y_45scale)) / (x_45scale / b)))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 1.7e-127) {
		tmp = (a_m / x_45_scale) * (a_m * (((b / (y_45_scale / -4.0)) / x_45_scale) / (y_45_scale / b)));
	} else if (a_m <= 2.9e+153) {
		tmp = (b / (y_45_scale * x_45_scale)) * ((-4.0 * (b * (a_m * a_m))) / (y_45_scale * x_45_scale));
	} else {
		tmp = (a_m / x_45_scale) * (b * ((a_m * ((-4.0 / y_45_scale) / y_45_scale)) / (x_45_scale / b)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 1.7e-127:
		tmp = (a_m / x_45_scale) * (a_m * (((b / (y_45_scale / -4.0)) / x_45_scale) / (y_45_scale / b)))
	elif a_m <= 2.9e+153:
		tmp = (b / (y_45_scale * x_45_scale)) * ((-4.0 * (b * (a_m * a_m))) / (y_45_scale * x_45_scale))
	else:
		tmp = (a_m / x_45_scale) * (b * ((a_m * ((-4.0 / y_45_scale) / y_45_scale)) / (x_45_scale / b)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 1.7e-127)
		tmp = Float64(Float64(a_m / x_45_scale) * Float64(a_m * Float64(Float64(Float64(b / Float64(y_45_scale / -4.0)) / x_45_scale) / Float64(y_45_scale / b))));
	elseif (a_m <= 2.9e+153)
		tmp = Float64(Float64(b / Float64(y_45_scale * x_45_scale)) * Float64(Float64(-4.0 * Float64(b * Float64(a_m * a_m))) / Float64(y_45_scale * x_45_scale)));
	else
		tmp = Float64(Float64(a_m / x_45_scale) * Float64(b * Float64(Float64(a_m * Float64(Float64(-4.0 / y_45_scale) / y_45_scale)) / Float64(x_45_scale / b))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 1.7e-127)
		tmp = (a_m / x_45_scale) * (a_m * (((b / (y_45_scale / -4.0)) / x_45_scale) / (y_45_scale / b)));
	elseif (a_m <= 2.9e+153)
		tmp = (b / (y_45_scale * x_45_scale)) * ((-4.0 * (b * (a_m * a_m))) / (y_45_scale * x_45_scale));
	else
		tmp = (a_m / x_45_scale) * (b * ((a_m * ((-4.0 / y_45_scale) / y_45_scale)) / (x_45_scale / b)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 1.7e-127], N[(N[(a$95$m / x$45$scale), $MachinePrecision] * N[(a$95$m * N[(N[(N[(b / N[(y$45$scale / -4.0), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 2.9e+153], N[(N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * N[(b * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / x$45$scale), $MachinePrecision] * N[(b * N[(N[(a$95$m * N[(N[(-4.0 / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.6999999999999999e-127

   1. Initial program 29.7%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

      17. *-lowering-*.f64 45.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

   5. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 57.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

   7. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale}\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{1}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \left(\frac{x-scale}{b}\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      12. /-lowering-/.f64 75.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \mathsf{/.f64}\left(x-scale, b\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. Applied egg-rr 75.5%

      \[\leadsto \left(\color{blue}{\frac{\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}}{y-scale}} \cdot a\right) \cdot \frac{a}{x-scale} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{-4}{y-scale} \cdot b}{\frac{x-scale}{b}}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{-4}{y-scale} \cdot b}{x-scale \cdot \frac{1}{b}}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{-4}{y-scale} \cdot b}{x-scale}}{\frac{1}{b}}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{-4}{y-scale} \cdot b}{x-scale}}{y-scale \cdot \frac{1}{b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       5. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{-4}{y-scale} \cdot b}{x-scale}}{\frac{y-scale}{b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot b}{x-scale}\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale} \cdot b\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot \frac{-4}{y-scale}\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot \frac{1}{\frac{y-scale}{-4}}\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       10. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{b}{\frac{y-scale}{-4}}\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \left(\frac{y-scale}{-4}\right)\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(y-scale, -4\right)\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       13. /-lowering-/.f64 81.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(y-scale, -4\right)\right), x-scale\right), \mathsf{/.f64}\left(y-scale, b\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   11. Applied egg-rr 81.5%

       \[\leadsto \left(\color{blue}{\frac{\frac{\frac{b}{\frac{y-scale}{-4}}}{x-scale}}{\frac{y-scale}{b}}} \cdot a\right) \cdot \frac{a}{x-scale} \]

   if 1.6999999999999999e-127 < a < 2.90000000000000002e153

   1. Initial program 35.4%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

      17. *-lowering-*.f64 66.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 70.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

   7. Applied egg-rr 70.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot \color{blue}{\left(a \cdot \frac{a}{x-scale}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot \frac{a \cdot a}{\color{blue}{x-scale}} \]

      3. div-inv N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right) \cdot \frac{\color{blue}{a \cdot a}}{x-scale} \]

      4. clear-num N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \frac{a \cdot \color{blue}{a}}{x-scale} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \color{blue}{\left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]

      6. frac-times N/A

         \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot x-scale}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot \left(b \cdot \left(a \cdot a\right)\right)}{\color{blue}{x-scale} \cdot x-scale} \]

      8. times-frac N/A

         \[\leadsto \frac{-4 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right) \cdot -4}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot \left(x-scale \cdot x-scale\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{b \cdot \left(\left(b \cdot \left(a \cdot a\right)\right) \cdot -4\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot \left(x-scale \cdot x-scale\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{b \cdot \left(\left(b \cdot \left(a \cdot a\right)\right) \cdot -4\right)}{\left(x-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}} \]

      12. swap-sqr N/A

          \[\leadsto \frac{b \cdot \left(\left(b \cdot \left(a \cdot a\right)\right) \cdot -4\right)}{\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}} \]

      13. times-frac N/A

          \[\leadsto \frac{b}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot -4}{x-scale \cdot y-scale}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{b}{x-scale \cdot y-scale}\right), \color{blue}{\left(\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot -4}{x-scale \cdot y-scale}\right)}\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \left(x-scale \cdot y-scale\right)\right), \left(\frac{\color{blue}{\left(b \cdot \left(a \cdot a\right)\right) \cdot -4}}{x-scale \cdot y-scale}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \left(y-scale \cdot x-scale\right)\right), \left(\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{-4}}{x-scale \cdot y-scale}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(y-scale, x-scale\right)\right), \left(\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{-4}}{x-scale \cdot y-scale}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(y-scale, x-scale\right)\right), \mathsf{/.f64}\left(\left(\left(b \cdot \left(a \cdot a\right)\right) \cdot -4\right), \color{blue}{\left(x-scale \cdot y-scale\right)}\right)\right) \]

   9. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\frac{b}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot \left(b \cdot \left(a \cdot a\right)\right)}{y-scale \cdot x-scale}} \]

   if 2.90000000000000002e153 < a

   1. Initial program 0.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

      17. *-lowering-*.f64 47.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

   5. Simplified 47.9%

      \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 69.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

   7. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale} \cdot a}{\frac{x-scale}{b \cdot b}}\right), \mathsf{/.f64}\left(\color{blue}{a}, x-scale\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale} \cdot a}{\frac{\frac{x-scale}{b}}{b}}\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale} \cdot a}{\frac{x-scale}{b}} \cdot b\right), \mathsf{/.f64}\left(\color{blue}{a}, x-scale\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale} \cdot a}{\frac{x-scale}{b}}\right), b\right), \mathsf{/.f64}\left(\color{blue}{a}, x-scale\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale}\right), a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale}\right), y-scale\right), a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), y-scale\right), a\right), \left(\frac{x-scale}{b}\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      10. /-lowering-/.f64 86.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), y-scale\right), a\right), \mathsf{/.f64}\left(x-scale, b\right)\right), b\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{\left(\frac{\frac{\frac{-4}{y-scale}}{y-scale} \cdot a}{\frac{x-scale}{b}} \cdot b\right)} \cdot \frac{a}{x-scale} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.7 \cdot 10^{-127}:\\ \;\;\;\;\frac{a}{x-scale} \cdot \left(a \cdot \frac{\frac{\frac{b}{\frac{y-scale}{-4}}}{x-scale}}{\frac{y-scale}{b}}\right)\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{y-scale \cdot x-scale} \cdot \frac{-4 \cdot \left(b \cdot \left(a \cdot a\right)\right)}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{x-scale} \cdot \left(b \cdot \frac{a \cdot \frac{\frac{-4}{y-scale}}{y-scale}}{\frac{x-scale}{b}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 76.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-158}:\\ \;\;\;\;\left(b \cdot \left(a\_m \cdot a\_m\right)\right) \cdot \left(b \cdot \frac{-4}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{x-scale} \cdot \left(-4 \cdot \frac{\frac{\frac{a\_m \cdot \left(b \cdot b\right)}{y-scale}}{x-scale}}{y-scale}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= b 2.8e-158)
   (*
    (* b (* a_m a_m))
    (* b (/ -4.0 (* x-scale (* y-scale (* y-scale x-scale))))))
   (*
    (/ a_m x-scale)
    (* -4.0 (/ (/ (/ (* a_m (* b b)) y-scale) x-scale) y-scale)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 2.8e-158) {
		tmp = (b * (a_m * a_m)) * (b * (-4.0 / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale)))));
	} else {
		tmp = (a_m / x_45_scale) * (-4.0 * ((((a_m * (b * b)) / y_45_scale) / x_45_scale) / y_45_scale));
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 2.8d-158) then
        tmp = (b * (a_m * a_m)) * (b * ((-4.0d0) / (x_45scale * (y_45scale * (y_45scale * x_45scale)))))
    else
        tmp = (a_m / x_45scale) * ((-4.0d0) * ((((a_m * (b * b)) / y_45scale) / x_45scale) / y_45scale))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 2.8e-158) {
		tmp = (b * (a_m * a_m)) * (b * (-4.0 / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale)))));
	} else {
		tmp = (a_m / x_45_scale) * (-4.0 * ((((a_m * (b * b)) / y_45_scale) / x_45_scale) / y_45_scale));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 2.8e-158:
		tmp = (b * (a_m * a_m)) * (b * (-4.0 / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale)))))
	else:
		tmp = (a_m / x_45_scale) * (-4.0 * ((((a_m * (b * b)) / y_45_scale) / x_45_scale) / y_45_scale))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 2.8e-158)
		tmp = Float64(Float64(b * Float64(a_m * a_m)) * Float64(b * Float64(-4.0 / Float64(x_45_scale * Float64(y_45_scale * Float64(y_45_scale * x_45_scale))))));
	else
		tmp = Float64(Float64(a_m / x_45_scale) * Float64(-4.0 * Float64(Float64(Float64(Float64(a_m * Float64(b * b)) / y_45_scale) / x_45_scale) / y_45_scale)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 2.8e-158)
		tmp = (b * (a_m * a_m)) * (b * (-4.0 / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale)))));
	else
		tmp = (a_m / x_45_scale) * (-4.0 * ((((a_m * (b * b)) / y_45_scale) / x_45_scale) / y_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 2.8e-158], N[(N[(b * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b * N[(-4.0 / N[(x$45$scale * N[(y$45$scale * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / x$45$scale), $MachinePrecision] * N[(-4.0 * N[(N[(N[(N[(a$95$m * N[(b * b), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.80000000000000002e-158

   1. Initial program 29.7%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

      17. *-lowering-*.f64 52.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

   5. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 64.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

   7. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot \color{blue}{\left(a \cdot \frac{a}{x-scale}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot \frac{a \cdot a}{\color{blue}{x-scale}} \]

      3. div-inv N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right) \cdot \frac{\color{blue}{a \cdot a}}{x-scale} \]

      4. clear-num N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \frac{a \cdot \color{blue}{a}}{x-scale} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \color{blue}{\left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]

      6. frac-times N/A

         \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot x-scale}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot \left(b \cdot \left(a \cdot a\right)\right)}{\color{blue}{x-scale} \cdot x-scale} \]

      8. times-frac N/A

         \[\leadsto \frac{-4 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right) \cdot -4}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot \left(x-scale \cdot x-scale\right)} \]

      10. associate-/l* N/A

          \[\leadsto \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\frac{-4}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(b \cdot \left(a \cdot a\right)\right) \cdot b\right) \cdot \frac{\color{blue}{-4}}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \]

      12. associate-*l* N/A

          \[\leadsto \left(b \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot \frac{-4}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b \cdot \left(a \cdot a\right)\right), \color{blue}{\left(b \cdot \frac{-4}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(a \cdot a\right)\right), \left(\color{blue}{b} \cdot \frac{-4}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right), \left(b \cdot \frac{-4}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{-4}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)}\right)\right) \]

   9. Applied egg-rr 66.1%

      \[\leadsto \color{blue}{\left(b \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot \frac{-4}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\right)} \]

   if 2.80000000000000002e-158 < b

   1. Initial program 20.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

      10. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

      17. *-lowering-*.f64 47.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

   5. Simplified 47.0%

      \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

      2. associate-/l* N/A

         \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 57.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

   7. Applied egg-rr 57.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale}\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{1}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \left(\frac{x-scale}{b}\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

      12. /-lowering-/.f64 74.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \mathsf{/.f64}\left(x-scale, b\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. Applied egg-rr 74.4%

      \[\leadsto \left(\color{blue}{\frac{\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}}{y-scale}} \cdot a\right) \cdot \frac{a}{x-scale} \]

   10. Taylor expanded in y-scale around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \frac{a \cdot {b}^{2}}{x-scale \cdot {y-scale}^{2}}\right)}, \mathsf{/.f64}\left(a, x-scale\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{a \cdot {b}^{2}}{x-scale \cdot {y-scale}^{2}}\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, x-scale\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{a \cdot {b}^{2}}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{a \cdot {b}^{2}}{\left(x-scale \cdot y-scale\right) \cdot y-scale}\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{\frac{a \cdot {b}^{2}}{x-scale \cdot y-scale}}{y-scale}\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\frac{a \cdot {b}^{2}}{x-scale \cdot y-scale}\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       6. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\frac{\frac{a \cdot {b}^{2}}{y-scale}}{x-scale}\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{a \cdot {b}^{2}}{y-scale}\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot {b}^{2}\right), y-scale\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), y-scale\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), y-scale\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

       11. *-lowering-*.f64 70.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), y-scale\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   12. Simplified 70.3%

       \[\leadsto \color{blue}{\left(-4 \cdot \frac{\frac{\frac{a \cdot \left(b \cdot b\right)}{y-scale}}{x-scale}}{y-scale}\right)} \cdot \frac{a}{x-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-158}:\\ \;\;\;\;\left(b \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot \frac{-4}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{x-scale} \cdot \left(-4 \cdot \frac{\frac{\frac{a \cdot \left(b \cdot b\right)}{y-scale}}{x-scale}}{y-scale}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 99.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \frac{a\_m}{x-scale} \cdot \left(a\_m \cdot \frac{\frac{\frac{b}{\frac{y-scale}{-4}}}{x-scale}}{\frac{y-scale}{b}}\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  (/ a_m x-scale)
  (* a_m (/ (/ (/ b (/ y-scale -4.0)) x-scale) (/ y-scale b)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a_m / x_45_scale) * (a_m * (((b / (y_45_scale / -4.0)) / x_45_scale) / (y_45_scale / b)));
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (a_m / x_45scale) * (a_m * (((b / (y_45scale / (-4.0d0))) / x_45scale) / (y_45scale / b)))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a_m / x_45_scale) * (a_m * (((b / (y_45_scale / -4.0)) / x_45_scale) / (y_45_scale / b)));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return (a_m / x_45_scale) * (a_m * (((b / (y_45_scale / -4.0)) / x_45_scale) / (y_45_scale / b)))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(a_m / x_45_scale) * Float64(a_m * Float64(Float64(Float64(b / Float64(y_45_scale / -4.0)) / x_45_scale) / Float64(y_45_scale / b))))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = (a_m / x_45_scale) * (a_m * (((b / (y_45_scale / -4.0)) / x_45_scale) / (y_45_scale / b)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(a$95$m / x$45$scale), $MachinePrecision] * N[(a$95$m * N[(N[(N[(b / N[(y$45$scale / -4.0), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.9%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

   17. *-lowering-*.f64 51.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

5. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

   2. associate-/l* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   13. /-lowering-/.f64 62.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

7. Applied egg-rr 62.0%

   \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
8. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale}\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{1}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   10. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \left(\frac{x-scale}{b}\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   12. /-lowering-/.f64 77.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \mathsf{/.f64}\left(x-scale, b\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

9. Applied egg-rr 77.2%

   \[\leadsto \left(\color{blue}{\frac{\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}}{y-scale}} \cdot a\right) \cdot \frac{a}{x-scale} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{-4}{y-scale} \cdot b}{\frac{x-scale}{b}}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    2. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{-4}{y-scale} \cdot b}{x-scale \cdot \frac{1}{b}}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    3. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{\frac{-4}{y-scale} \cdot b}{x-scale}}{\frac{1}{b}}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    4. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{-4}{y-scale} \cdot b}{x-scale}}{y-scale \cdot \frac{1}{b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    5. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{-4}{y-scale} \cdot b}{x-scale}}{\frac{y-scale}{b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot b}{x-scale}\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale} \cdot b\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot \frac{-4}{y-scale}\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot \frac{1}{\frac{y-scale}{-4}}\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    10. un-div-inv N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{b}{\frac{y-scale}{-4}}\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \left(\frac{y-scale}{-4}\right)\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(y-scale, -4\right)\right), x-scale\right), \left(\frac{y-scale}{b}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    13. /-lowering-/.f64 84.7%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(y-scale, -4\right)\right), x-scale\right), \mathsf{/.f64}\left(y-scale, b\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

11. Applied egg-rr 84.7%

    \[\leadsto \left(\color{blue}{\frac{\frac{\frac{b}{\frac{y-scale}{-4}}}{x-scale}}{\frac{y-scale}{b}}} \cdot a\right) \cdot \frac{a}{x-scale} \]

12. Final simplification 84.7%

    \[\leadsto \frac{a}{x-scale} \cdot \left(a \cdot \frac{\frac{\frac{b}{\frac{y-scale}{-4}}}{x-scale}}{\frac{y-scale}{b}}\right) \]
13. Add Preprocessing

Alternative 5: 82.9% accurate, 99.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \frac{a\_m}{x-scale} \cdot \left(a\_m \cdot \left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{\frac{y-scale}{-4}}\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  (/ a_m x-scale)
  (* a_m (* (/ b (* y-scale x-scale)) (/ b (/ y-scale -4.0))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a_m / x_45_scale) * (a_m * ((b / (y_45_scale * x_45_scale)) * (b / (y_45_scale / -4.0))));
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (a_m / x_45scale) * (a_m * ((b / (y_45scale * x_45scale)) * (b / (y_45scale / (-4.0d0)))))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a_m / x_45_scale) * (a_m * ((b / (y_45_scale * x_45_scale)) * (b / (y_45_scale / -4.0))));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return (a_m / x_45_scale) * (a_m * ((b / (y_45_scale * x_45_scale)) * (b / (y_45_scale / -4.0))))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(a_m / x_45_scale) * Float64(a_m * Float64(Float64(b / Float64(y_45_scale * x_45_scale)) * Float64(b / Float64(y_45_scale / -4.0)))))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = (a_m / x_45_scale) * (a_m * ((b / (y_45_scale * x_45_scale)) * (b / (y_45_scale / -4.0))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(a$95$m / x$45$scale), $MachinePrecision] * N[(a$95$m * N[(N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / N[(y$45$scale / -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.9%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

   17. *-lowering-*.f64 51.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

5. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

   2. associate-/l* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   13. /-lowering-/.f64 62.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

7. Applied egg-rr 62.0%

   \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
8. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale}\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{1}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   10. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \left(\frac{x-scale}{b}\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   12. /-lowering-/.f64 77.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \mathsf{/.f64}\left(x-scale, b\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

9. Applied egg-rr 77.2%

   \[\leadsto \left(\color{blue}{\frac{\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}}{y-scale}} \cdot a\right) \cdot \frac{a}{x-scale} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{-4}{y-scale} \cdot b}{\frac{x-scale}{b}}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    2. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot b}{y-scale \cdot \frac{x-scale}{b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    3. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale} \cdot b\right) \cdot \frac{1}{y-scale \cdot \frac{x-scale}{b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    4. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale} \cdot b\right) \cdot \frac{1}{\frac{y-scale \cdot x-scale}{b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    5. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale} \cdot b\right) \cdot \frac{b}{y-scale \cdot x-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale} \cdot b\right), \left(\frac{b}{y-scale \cdot x-scale}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot \frac{-4}{y-scale}\right), \left(\frac{b}{y-scale \cdot x-scale}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    8. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot \frac{1}{\frac{y-scale}{-4}}\right), \left(\frac{b}{y-scale \cdot x-scale}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    9. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{b}{\frac{y-scale}{-4}}\right), \left(\frac{b}{y-scale \cdot x-scale}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \left(\frac{y-scale}{-4}\right)\right), \left(\frac{b}{y-scale \cdot x-scale}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(y-scale, -4\right)\right), \left(\frac{b}{y-scale \cdot x-scale}\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(y-scale, -4\right)\right), \mathsf{/.f64}\left(b, \left(y-scale \cdot x-scale\right)\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    13. *-lowering-*.f64 82.4%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(y-scale, -4\right)\right), \mathsf{/.f64}\left(b, \mathsf{*.f64}\left(y-scale, x-scale\right)\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

11. Applied egg-rr 82.4%

    \[\leadsto \left(\color{blue}{\left(\frac{b}{\frac{y-scale}{-4}} \cdot \frac{b}{y-scale \cdot x-scale}\right)} \cdot a\right) \cdot \frac{a}{x-scale} \]

12. Final simplification 82.4%

    \[\leadsto \frac{a}{x-scale} \cdot \left(a \cdot \left(\frac{b}{y-scale \cdot x-scale} \cdot \frac{b}{\frac{y-scale}{-4}}\right)\right) \]
13. Add Preprocessing

Alternative 6: 75.4% accurate, 99.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \frac{a\_m}{x-scale} \cdot \left(a\_m \cdot \left(b \cdot \left(b \cdot \frac{-4}{y-scale \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  (/ a_m x-scale)
  (* a_m (* b (* b (/ -4.0 (* y-scale (* y-scale x-scale))))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a_m / x_45_scale) * (a_m * (b * (b * (-4.0 / (y_45_scale * (y_45_scale * x_45_scale))))));
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (a_m / x_45scale) * (a_m * (b * (b * ((-4.0d0) / (y_45scale * (y_45scale * x_45scale))))))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a_m / x_45_scale) * (a_m * (b * (b * (-4.0 / (y_45_scale * (y_45_scale * x_45_scale))))));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return (a_m / x_45_scale) * (a_m * (b * (b * (-4.0 / (y_45_scale * (y_45_scale * x_45_scale))))))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(a_m / x_45_scale) * Float64(a_m * Float64(b * Float64(b * Float64(-4.0 / Float64(y_45_scale * Float64(y_45_scale * x_45_scale)))))))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = (a_m / x_45_scale) * (a_m * (b * (b * (-4.0 / (y_45_scale * (y_45_scale * x_45_scale))))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(a$95$m / x$45$scale), $MachinePrecision] * N[(a$95$m * N[(b * N[(b * N[(-4.0 / N[(y$45$scale * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.9%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

   17. *-lowering-*.f64 51.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

5. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

   2. associate-/l* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   13. /-lowering-/.f64 62.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

7. Applied egg-rr 62.0%

   \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
8. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{x-scale} \cdot \left(b \cdot b\right)\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{x-scale} \cdot b\right) \cdot b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{x-scale} \cdot b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{x-scale}\right), b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   5. associate-/l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right), b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{\left(y-scale \cdot y-scale\right) \cdot x-scale}\right), b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right)\right), b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)\right), b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)\right), b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, \left(x-scale \cdot y-scale\right)\right)\right), b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, \left(y-scale \cdot x-scale\right)\right)\right), b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   12. *-lowering-*.f64 73.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, \mathsf{*.f64}\left(y-scale, x-scale\right)\right)\right), b\right), b\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

9. Applied egg-rr 73.8%

   \[\leadsto \left(\color{blue}{\left(\left(\frac{-4}{y-scale \cdot \left(y-scale \cdot x-scale\right)} \cdot b\right) \cdot b\right)} \cdot a\right) \cdot \frac{a}{x-scale} \]

10. Final simplification 73.8%

    \[\leadsto \frac{a}{x-scale} \cdot \left(a \cdot \left(b \cdot \left(b \cdot \frac{-4}{y-scale \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \]
11. Add Preprocessing

Alternative 7: 72.5% accurate, 99.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \frac{a\_m}{x-scale} \cdot \left(-4 \cdot \frac{\frac{\frac{a\_m \cdot \left(b \cdot b\right)}{y-scale}}{x-scale}}{y-scale}\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  (/ a_m x-scale)
  (* -4.0 (/ (/ (/ (* a_m (* b b)) y-scale) x-scale) y-scale))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a_m / x_45_scale) * (-4.0 * ((((a_m * (b * b)) / y_45_scale) / x_45_scale) / y_45_scale));
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (a_m / x_45scale) * ((-4.0d0) * ((((a_m * (b * b)) / y_45scale) / x_45scale) / y_45scale))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a_m / x_45_scale) * (-4.0 * ((((a_m * (b * b)) / y_45_scale) / x_45_scale) / y_45_scale));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return (a_m / x_45_scale) * (-4.0 * ((((a_m * (b * b)) / y_45_scale) / x_45_scale) / y_45_scale))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(a_m / x_45_scale) * Float64(-4.0 * Float64(Float64(Float64(Float64(a_m * Float64(b * b)) / y_45_scale) / x_45_scale) / y_45_scale)))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = (a_m / x_45_scale) * (-4.0 * ((((a_m * (b * b)) / y_45_scale) / x_45_scale) / y_45_scale));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(a$95$m / x$45$scale), $MachinePrecision] * N[(-4.0 * N[(N[(N[(N[(a$95$m * N[(b * b), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.9%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

   17. *-lowering-*.f64 51.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

5. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{b \cdot b}{x-scale} \cdot \frac{a \cdot a}{x-scale}\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \color{blue}{\frac{a \cdot a}{x-scale}} \]

   2. associate-/l* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{a}{x-scale}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right) \cdot \color{blue}{\frac{a}{x-scale}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right) \cdot a\right), \color{blue}{\left(\frac{a}{x-scale}\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \left(\frac{\color{blue}{a}}{x-scale}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}}\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{b \cdot b}\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(b \cdot b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \left(\frac{a}{x-scale}\right)\right) \]

   13. /-lowering-/.f64 62.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right), \mathsf{/.f64}\left(a, \color{blue}{x-scale}\right)\right) \]

7. Applied egg-rr 62.0%

   \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{b \cdot b}} \cdot a\right) \cdot \frac{a}{x-scale}} \]
8. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{1}{\frac{x-scale}{b \cdot b}}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}}{y-scale}\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale} \cdot \frac{b \cdot b}{x-scale}\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale}\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b \cdot b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{b}{x-scale}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(b \cdot \frac{1}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   10. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \left(\frac{b}{\frac{x-scale}{b}}\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \left(\frac{x-scale}{b}\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

   12. /-lowering-/.f64 77.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, y-scale\right), \mathsf{/.f64}\left(b, \mathsf{/.f64}\left(x-scale, b\right)\right)\right), y-scale\right), a\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

9. Applied egg-rr 77.2%

   \[\leadsto \left(\color{blue}{\frac{\frac{-4}{y-scale} \cdot \frac{b}{\frac{x-scale}{b}}}{y-scale}} \cdot a\right) \cdot \frac{a}{x-scale} \]

10. Taylor expanded in y-scale around 0

    \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \frac{a \cdot {b}^{2}}{x-scale \cdot {y-scale}^{2}}\right)}, \mathsf{/.f64}\left(a, x-scale\right)\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{a \cdot {b}^{2}}{x-scale \cdot {y-scale}^{2}}\right)\right), \mathsf{/.f64}\left(\color{blue}{a}, x-scale\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{a \cdot {b}^{2}}{x-scale \cdot \left(y-scale \cdot y-scale\right)}\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{a \cdot {b}^{2}}{\left(x-scale \cdot y-scale\right) \cdot y-scale}\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    4. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(\frac{\frac{a \cdot {b}^{2}}{x-scale \cdot y-scale}}{y-scale}\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\frac{a \cdot {b}^{2}}{x-scale \cdot y-scale}\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    6. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left(\frac{\frac{a \cdot {b}^{2}}{y-scale}}{x-scale}\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{a \cdot {b}^{2}}{y-scale}\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot {b}^{2}\right), y-scale\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2}\right)\right), y-scale\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(b \cdot b\right)\right), y-scale\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

    11. *-lowering-*.f64 70.5%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right), y-scale\right), x-scale\right), y-scale\right)\right), \mathsf{/.f64}\left(a, x-scale\right)\right) \]

12. Simplified 70.5%

    \[\leadsto \color{blue}{\left(-4 \cdot \frac{\frac{\frac{a \cdot \left(b \cdot b\right)}{y-scale}}{x-scale}}{y-scale}\right)} \cdot \frac{a}{x-scale} \]

13. Final simplification 70.5%

    \[\leadsto \frac{a}{x-scale} \cdot \left(-4 \cdot \frac{\frac{\frac{a \cdot \left(b \cdot b\right)}{y-scale}}{x-scale}}{y-scale}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale)) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale))))