Simplification of discriminant from scale-rotated-ellipse

Average Error: 40.5 → 7.3

Time: 1.6min

Precision: binary64

Cost: 1880

Math

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

↓

\[\begin{array}{l} t_0 := a \cdot \frac{b}{y-scale}\\ t_1 := -4 \cdot \frac{t_0}{x-scale \cdot \frac{\frac{y-scale}{b}}{\frac{a}{x-scale}}}\\ t_2 := \frac{\left(a \cdot b\right) \cdot \left(-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)}{x-scale \cdot y-scale}\\ \mathbf{if}\;x-scale \leq -2.5 \cdot 10^{+163}:\\ \;\;\;\;\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}} \cdot \frac{a}{x-scale \cdot \left(-\frac{y-scale}{b}\right)}\right) \cdot 4\\ \mathbf{elif}\;x-scale \leq -6.6 \cdot 10^{+66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq -1.45 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -1.95 \cdot 10^{-253}:\\ \;\;\;\;\frac{t_0}{x-scale} \cdot \frac{-4}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 2 \cdot 10^{-205}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq 7.8 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{\frac{-4}{\frac{x-scale \cdot y-scale}{a \cdot b}}}{x-scale \cdot y-scale}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :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))))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (/ b y-scale)))
        (t_1 (* -4.0 (/ t_0 (* x-scale (/ (/ y-scale b) (/ a x-scale))))))
        (t_2
         (/
          (* (* a b) (* -4.0 (/ (* a b) (* x-scale y-scale))))
          (* x-scale y-scale))))
   (if (<= x-scale -2.5e+163)
     (*
      (* (/ (/ a x-scale) (/ y-scale b)) (/ a (* x-scale (- (/ y-scale b)))))
      4.0)
     (if (<= x-scale -6.6e+66)
       t_2
       (if (<= x-scale -1.45e-143)
         t_1
         (if (<= x-scale -1.95e-253)
           (* (/ t_0 x-scale) (/ -4.0 (* (/ x-scale b) (/ y-scale a))))
           (if (<= x-scale 2e-205)
             t_2
             (if (<= x-scale 7.8e+146)
               t_1
               (*
                (* a b)
                (/
                 (/ -4.0 (/ (* x-scale y-scale) (* a b)))
                 (* x-scale y-scale)))))))))))

C

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

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / y_45_scale);
	double t_1 = -4.0 * (t_0 / (x_45_scale * ((y_45_scale / b) / (a / x_45_scale))));
	double t_2 = ((a * b) * (-4.0 * ((a * b) / (x_45_scale * y_45_scale)))) / (x_45_scale * y_45_scale);
	double tmp;
	if (x_45_scale <= -2.5e+163) {
		tmp = (((a / x_45_scale) / (y_45_scale / b)) * (a / (x_45_scale * -(y_45_scale / b)))) * 4.0;
	} else if (x_45_scale <= -6.6e+66) {
		tmp = t_2;
	} else if (x_45_scale <= -1.45e-143) {
		tmp = t_1;
	} else if (x_45_scale <= -1.95e-253) {
		tmp = (t_0 / x_45_scale) * (-4.0 / ((x_45_scale / b) * (y_45_scale / a)));
	} else if (x_45_scale <= 2e-205) {
		tmp = t_2;
	} else if (x_45_scale <= 7.8e+146) {
		tmp = t_1;
	} else {
		tmp = (a * b) * ((-4.0 / ((x_45_scale * y_45_scale) / (a * b))) / (x_45_scale * y_45_scale));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / y_45_scale);
	double t_1 = -4.0 * (t_0 / (x_45_scale * ((y_45_scale / b) / (a / x_45_scale))));
	double t_2 = ((a * b) * (-4.0 * ((a * b) / (x_45_scale * y_45_scale)))) / (x_45_scale * y_45_scale);
	double tmp;
	if (x_45_scale <= -2.5e+163) {
		tmp = (((a / x_45_scale) / (y_45_scale / b)) * (a / (x_45_scale * -(y_45_scale / b)))) * 4.0;
	} else if (x_45_scale <= -6.6e+66) {
		tmp = t_2;
	} else if (x_45_scale <= -1.45e-143) {
		tmp = t_1;
	} else if (x_45_scale <= -1.95e-253) {
		tmp = (t_0 / x_45_scale) * (-4.0 / ((x_45_scale / b) * (y_45_scale / a)));
	} else if (x_45_scale <= 2e-205) {
		tmp = t_2;
	} else if (x_45_scale <= 7.8e+146) {
		tmp = t_1;
	} else {
		tmp = (a * b) * ((-4.0 / ((x_45_scale * y_45_scale) / (a * b))) / (x_45_scale * y_45_scale));
	}
	return tmp;
}

Python

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

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * (b / y_45_scale)
	t_1 = -4.0 * (t_0 / (x_45_scale * ((y_45_scale / b) / (a / x_45_scale))))
	t_2 = ((a * b) * (-4.0 * ((a * b) / (x_45_scale * y_45_scale)))) / (x_45_scale * y_45_scale)
	tmp = 0
	if x_45_scale <= -2.5e+163:
		tmp = (((a / x_45_scale) / (y_45_scale / b)) * (a / (x_45_scale * -(y_45_scale / b)))) * 4.0
	elif x_45_scale <= -6.6e+66:
		tmp = t_2
	elif x_45_scale <= -1.45e-143:
		tmp = t_1
	elif x_45_scale <= -1.95e-253:
		tmp = (t_0 / x_45_scale) * (-4.0 / ((x_45_scale / b) * (y_45_scale / a)))
	elif x_45_scale <= 2e-205:
		tmp = t_2
	elif x_45_scale <= 7.8e+146:
		tmp = t_1
	else:
		tmp = (a * b) * ((-4.0 / ((x_45_scale * y_45_scale) / (a * b))) / (x_45_scale * y_45_scale))
	return tmp

Julia

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

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(b / y_45_scale))
	t_1 = Float64(-4.0 * Float64(t_0 / Float64(x_45_scale * Float64(Float64(y_45_scale / b) / Float64(a / x_45_scale)))))
	t_2 = Float64(Float64(Float64(a * b) * Float64(-4.0 * Float64(Float64(a * b) / Float64(x_45_scale * y_45_scale)))) / Float64(x_45_scale * y_45_scale))
	tmp = 0.0
	if (x_45_scale <= -2.5e+163)
		tmp = Float64(Float64(Float64(Float64(a / x_45_scale) / Float64(y_45_scale / b)) * Float64(a / Float64(x_45_scale * Float64(-Float64(y_45_scale / b))))) * 4.0);
	elseif (x_45_scale <= -6.6e+66)
		tmp = t_2;
	elseif (x_45_scale <= -1.45e-143)
		tmp = t_1;
	elseif (x_45_scale <= -1.95e-253)
		tmp = Float64(Float64(t_0 / x_45_scale) * Float64(-4.0 / Float64(Float64(x_45_scale / b) * Float64(y_45_scale / a))));
	elseif (x_45_scale <= 2e-205)
		tmp = t_2;
	elseif (x_45_scale <= 7.8e+146)
		tmp = t_1;
	else
		tmp = Float64(Float64(a * b) * Float64(Float64(-4.0 / Float64(Float64(x_45_scale * y_45_scale) / Float64(a * b))) / Float64(x_45_scale * y_45_scale)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * (b / y_45_scale);
	t_1 = -4.0 * (t_0 / (x_45_scale * ((y_45_scale / b) / (a / x_45_scale))));
	t_2 = ((a * b) * (-4.0 * ((a * b) / (x_45_scale * y_45_scale)))) / (x_45_scale * y_45_scale);
	tmp = 0.0;
	if (x_45_scale <= -2.5e+163)
		tmp = (((a / x_45_scale) / (y_45_scale / b)) * (a / (x_45_scale * -(y_45_scale / b)))) * 4.0;
	elseif (x_45_scale <= -6.6e+66)
		tmp = t_2;
	elseif (x_45_scale <= -1.45e-143)
		tmp = t_1;
	elseif (x_45_scale <= -1.95e-253)
		tmp = (t_0 / x_45_scale) * (-4.0 / ((x_45_scale / b) * (y_45_scale / a)));
	elseif (x_45_scale <= 2e-205)
		tmp = t_2;
	elseif (x_45_scale <= 7.8e+146)
		tmp = t_1;
	else
		tmp = (a * b) * ((-4.0 / ((x_45_scale * y_45_scale) / (a * b))) / (x_45_scale * y_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(t$95$0 / N[(x$45$scale * N[(N[(y$45$scale / b), $MachinePrecision] / N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a * b), $MachinePrecision] * N[(-4.0 * N[(N[(a * b), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -2.5e+163], N[(N[(N[(N[(a / x$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision] * N[(a / N[(x$45$scale * (-N[(y$45$scale / b), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[x$45$scale, -6.6e+66], t$95$2, If[LessEqual[x$45$scale, -1.45e-143], t$95$1, If[LessEqual[x$45$scale, -1.95e-253], N[(N[(t$95$0 / x$45$scale), $MachinePrecision] * N[(-4.0 / N[(N[(x$45$scale / b), $MachinePrecision] * N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2e-205], t$95$2, If[LessEqual[x$45$scale, 7.8e+146], t$95$1, N[(N[(a * b), $MachinePrecision] * N[(N[(-4.0 / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x-scale < -2.5e163

   1. Initial program 37.6

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

   2. Simplified 40.9

      \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} \cdot \frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale \cdot x-scale} - \left(4 \cdot \frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}\right) \cdot \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}} \]
      Proof

      [Start] 37.6

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

   3. Taylor expanded in angle around 0 38.2

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

   4. Simplified 28.1

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

      [Start] 38.2	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 38.2	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 38.5	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 38.5	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 38.5	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 30.6	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 30.6	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 28.1	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 28.1	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Applied egg-rr 11.1

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

   6. Simplified 7.4

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

      [Start] 11.1	\[ -4 \cdot \frac{\frac{a}{x-scale} \cdot \left(-a\right)}{\frac{y-scale}{b} \cdot \left(\frac{y-scale}{b} \cdot \left(-x-scale\right)\right)} \]
      times-frac [=>] 7.4	\[ -4 \cdot \color{blue}{\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}} \cdot \frac{-a}{\frac{y-scale}{b} \cdot \left(-x-scale\right)}\right)} \]

   if -2.5e163 < x-scale < -6.6000000000000003e66 or -1.9499999999999999e-253 < x-scale < 2e-205

   1. Initial program 43.5

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

   2. Simplified 52.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}, \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} \cdot -4, \left(\frac{\sin \left(\frac{angle}{180} \cdot \pi\right)}{x-scale \cdot y-scale} \cdot \frac{\sin \left(\frac{angle}{180} \cdot \pi\right)}{x-scale \cdot y-scale}\right) \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(4 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\right)\right)} \]
      Proof

      [Start] 43.5	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 43.5	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 43.5	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 46.3

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

   4. Simplified 30.0

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

      [Start] 46.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 46.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-*r/ [=>] 46.3	\[ \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-/l* [=>] 46.3	\[ \color{blue}{\frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}}} \]
      unpow2 [=>] 46.3	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      unpow2 [=>] 46.3	\[ \frac{-4}{\frac{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      unswap-sqr [=>] 34.3	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 34.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{{b}^{2} \cdot {a}^{2}}}} \]
      unpow2 [=>] 34.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}} \]
      associate-*l* [=>] 30.0	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{b \cdot \left(b \cdot {a}^{2}\right)}}} \]
      unpow2 [=>] 30.0	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{b \cdot \left(b \cdot \color{blue}{\left(a \cdot a\right)}\right)}} \]

   5. Applied egg-rr 26.2

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

   6. Applied egg-rr 15.3

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

   7. Simplified 16.0

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

      [Start] 15.3	\[ \frac{-4}{x-scale \cdot \frac{y-scale}{b \cdot a}} \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \]
      *-commutative [=>] 15.3	\[ \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{-4}{x-scale \cdot \frac{y-scale}{b \cdot a}}} \]
      associate-*r/ [=>] 14.9	\[ \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{-4}{\color{blue}{\frac{x-scale \cdot y-scale}{b \cdot a}}} \]
      times-frac [=>] 16.0	\[ \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{-4}{\color{blue}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}} \]

   8. Applied egg-rr 11.0

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

   if -6.6000000000000003e66 < x-scale < -1.45e-143 or 2e-205 < x-scale < 7.8e146

   1. Initial program 40.8

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

   2. Simplified 45.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}, \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} \cdot -4, \frac{4 \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)}{x-scale \cdot \left(y-scale \cdot \frac{x-scale}{\cos \left(\frac{angle}{180} \cdot \pi\right)}\right)} \cdot \frac{\cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale}\right)} \]
      Proof

      [Start] 40.8	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 40.8	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 40.8	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 35.5

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

   4. Simplified 19.9

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

      [Start] 35.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 35.5	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot -4} \]
      times-frac [=>] 35.6	\[ \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 35.6	\[ \color{blue}{\frac{{a}^{2}}{{x-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 35.6	\[ \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 35.6	\[ \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 32.4	\[ \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 32.4	\[ \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 32.4	\[ \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot -4\right) \]
      times-frac [=>] 19.9	\[ \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot -4\right) \]

   5. Taylor expanded in a around 0 35.5

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

   6. Simplified 4.5

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

      [Start] 35.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 35.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
      times-frac [=>] 35.6	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 35.6	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 35.6	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      times-frac [=>] 32.4	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 32.4	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 32.4	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 19.8	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]
      swap-sqr [<=] 4.5	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      unpow2 [<=] 4.5	\[ -4 \cdot \color{blue}{{\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}} \]
      associate-*r/ [=>] 5.0	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{x-scale} \cdot b}{y-scale}\right)}}^{2} \]
      associate-/l* [=>] 4.5	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}}^{2} \]

   7. Applied egg-rr 4.6

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

   if -1.45e-143 < x-scale < -1.9499999999999999e-253

   1. Initial program 50.9

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

   2. Simplified 61.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}, \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} \cdot -4, \left(\frac{\sin \left(\frac{angle}{180} \cdot \pi\right)}{x-scale \cdot y-scale} \cdot \frac{\sin \left(\frac{angle}{180} \cdot \pi\right)}{x-scale \cdot y-scale}\right) \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(4 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\right)\right)} \]
      Proof

      [Start] 50.9	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 50.9	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 50.9	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 61.0

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

   4. Simplified 36.8

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

      [Start] 61.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 61.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-*r/ [=>] 61.0	\[ \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-/l* [=>] 61.0	\[ \color{blue}{\frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}}} \]
      unpow2 [=>] 61.0	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      unpow2 [=>] 61.0	\[ \frac{-4}{\frac{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      unswap-sqr [=>] 38.6	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 38.6	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{{b}^{2} \cdot {a}^{2}}}} \]
      unpow2 [=>] 38.6	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}} \]
      associate-*l* [=>] 36.8	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{b \cdot \left(b \cdot {a}^{2}\right)}}} \]
      unpow2 [=>] 36.8	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{b \cdot \left(b \cdot \color{blue}{\left(a \cdot a\right)}\right)}} \]

   5. Applied egg-rr 32.5

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

   6. Applied egg-rr 17.5

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

   7. Simplified 18.2

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

      [Start] 17.5	\[ \frac{-4}{x-scale \cdot \frac{y-scale}{b \cdot a}} \cdot \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \]
      *-commutative [=>] 17.5	\[ \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{-4}{x-scale \cdot \frac{y-scale}{b \cdot a}}} \]
      associate-*r/ [=>] 19.3	\[ \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{-4}{\color{blue}{\frac{x-scale \cdot y-scale}{b \cdot a}}} \]
      times-frac [=>] 18.2	\[ \left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{-4}{\color{blue}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}} \]

   8. Applied egg-rr 15.4

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

   if 7.8e146 < x-scale

   1. Initial program 35.7

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

   2. Simplified 42.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}, \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} \cdot -4, \left(\frac{\sin \left(\frac{angle}{180} \cdot \pi\right)}{x-scale \cdot y-scale} \cdot \frac{\sin \left(\frac{angle}{180} \cdot \pi\right)}{x-scale \cdot y-scale}\right) \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(4 \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)\right)\right)} \]
      Proof

      [Start] 35.7	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 35.7	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 35.7	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 36.0

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

   4. Simplified 23.3

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

      [Start] 36.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 36.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-*r/ [=>] 36.0	\[ \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-/l* [=>] 36.0	\[ \color{blue}{\frac{-4}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}}} \]
      unpow2 [=>] 36.0	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}}{{a}^{2} \cdot {b}^{2}}} \]
      unpow2 [=>] 36.0	\[ \frac{-4}{\frac{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      unswap-sqr [=>] 27.3	\[ \frac{-4}{\frac{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}{{a}^{2} \cdot {b}^{2}}} \]
      *-commutative [=>] 27.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{{b}^{2} \cdot {a}^{2}}}} \]
      unpow2 [=>] 27.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}} \]
      associate-*l* [=>] 23.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{b \cdot \left(b \cdot {a}^{2}\right)}}} \]
      unpow2 [=>] 23.3	\[ \frac{-4}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{b \cdot \left(b \cdot \color{blue}{\left(a \cdot a\right)}\right)}} \]

   5. Applied egg-rr 7.6

      \[\leadsto \color{blue}{\frac{\frac{-4}{\frac{y-scale \cdot x-scale}{b \cdot a}}}{y-scale \cdot x-scale} \cdot \left(b \cdot a\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -2.5 \cdot 10^{+163}:\\ \;\;\;\;\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}} \cdot \frac{a}{x-scale \cdot \left(-\frac{y-scale}{b}\right)}\right) \cdot 4\\ \mathbf{elif}\;x-scale \leq -6.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)}{x-scale \cdot y-scale}\\ \mathbf{elif}\;x-scale \leq -1.45 \cdot 10^{-143}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{b}{y-scale}}{x-scale \cdot \frac{\frac{y-scale}{b}}{\frac{a}{x-scale}}}\\ \mathbf{elif}\;x-scale \leq -1.95 \cdot 10^{-253}:\\ \;\;\;\;\frac{a \cdot \frac{b}{y-scale}}{x-scale} \cdot \frac{-4}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 2 \cdot 10^{-205}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)}{x-scale \cdot y-scale}\\ \mathbf{elif}\;x-scale \leq 7.8 \cdot 10^{+146}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{b}{y-scale}}{x-scale \cdot \frac{\frac{y-scale}{b}}{\frac{a}{x-scale}}}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{\frac{-4}{\frac{x-scale \cdot y-scale}{a \cdot b}}}{x-scale \cdot y-scale}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.0
Cost	7956

\[\begin{array}{l} t_0 := a \cdot \frac{b}{y-scale}\\ t_1 := -4 \cdot {\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}^{2}\\ \mathbf{if}\;x-scale \leq -4.4 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -4.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)}{x-scale \cdot y-scale}\\ \mathbf{elif}\;x-scale \leq -1.45 \cdot 10^{-143}:\\ \;\;\;\;-4 \cdot \frac{t_0}{x-scale \cdot \frac{\frac{y-scale}{b}}{\frac{a}{x-scale}}}\\ \mathbf{elif}\;x-scale \leq -1.3 \cdot 10^{-254}:\\ \;\;\;\;\frac{t_0}{x-scale} \cdot \frac{-4}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 1.75 \cdot 10^{-206}:\\ \;\;\;\;\frac{-4}{\left(x-scale \cdot y-scale\right) \cdot \frac{x-scale \cdot y-scale}{{\left(a \cdot b\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	6.5
Cost	7700

\[\begin{array}{l} t_0 := a \cdot \frac{b}{y-scale}\\ t_1 := \frac{\left(a \cdot b\right) \cdot \left(-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)}{x-scale \cdot y-scale}\\ t_2 := -4 \cdot {\left(\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}\right)}^{2}\\ \mathbf{if}\;x-scale \leq -6.8 \cdot 10^{+164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq -2.7 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -1.45 \cdot 10^{-143}:\\ \;\;\;\;-4 \cdot \frac{t_0}{x-scale \cdot \frac{\frac{y-scale}{b}}{\frac{a}{x-scale}}}\\ \mathbf{elif}\;x-scale \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\frac{t_0}{x-scale} \cdot \frac{-4}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 3.3 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	10.7
Cost	1748

\[\begin{array}{l} t_0 := -4 \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\right)\\ t_1 := \frac{a}{x-scale \cdot y-scale}\\ t_2 := -4 \cdot \left(b \cdot \frac{t_1}{\frac{x-scale}{\frac{a \cdot b}{y-scale}}}\right)\\ \mathbf{if}\;x-scale \leq -6.6 \cdot 10^{+163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -5.7 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq -2.45 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 1.96 \cdot 10^{-75}:\\ \;\;\;\;-4 \cdot \frac{b \cdot t_1}{\frac{x-scale}{a \cdot \frac{b}{y-scale}}}\\ \mathbf{elif}\;x-scale \leq 4.5 \cdot 10^{+247}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	10.7
Cost	1748

\[\begin{array}{l} t_0 := -4 \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\right)\\ t_1 := \frac{a}{x-scale \cdot y-scale}\\ t_2 := -4 \cdot \left(b \cdot \frac{t_1}{\frac{x-scale}{\frac{a \cdot b}{y-scale}}}\right)\\ \mathbf{if}\;x-scale \leq -2.85 \cdot 10^{+166}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale} \cdot \left(a \cdot \frac{a}{x-scale}\right)}{x-scale \cdot \frac{y-scale}{b}}\\ \mathbf{elif}\;x-scale \leq -4.8 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq -3.4 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 2 \cdot 10^{-79}:\\ \;\;\;\;-4 \cdot \frac{b \cdot t_1}{\frac{x-scale}{a \cdot \frac{b}{y-scale}}}\\ \mathbf{elif}\;x-scale \leq 5 \cdot 10^{+247}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	10.1
Cost	1616

\[\begin{array}{l} t_0 := \frac{a}{x-scale \cdot y-scale}\\ t_1 := -4 \cdot \left(b \cdot \frac{t_0}{\frac{x-scale}{\frac{a \cdot b}{y-scale}}}\right)\\ \mathbf{if}\;x-scale \leq -3 \cdot 10^{+164}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale}}{\frac{\frac{y-scale}{b}}{\frac{a}{x-scale}} \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;x-scale \leq -5.8 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;-4 \cdot \frac{b \cdot t_0}{\frac{x-scale}{a \cdot \frac{b}{y-scale}}}\\ \mathbf{elif}\;x-scale \leq 1.25 \cdot 10^{+247}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	9.9
Cost	1616

\[\begin{array}{l} t_0 := a \cdot \frac{-4 \cdot \frac{b}{x-scale \cdot y-scale}}{\frac{x-scale \cdot y-scale}{a \cdot b}}\\ \mathbf{if}\;x-scale \leq -2.6 \cdot 10^{+166}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale}}{\frac{\frac{y-scale}{b}}{\frac{a}{x-scale}} \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;x-scale \leq -5300:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 1.45 \cdot 10^{-75}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{a}{x-scale \cdot y-scale}}{\frac{x-scale}{a \cdot \frac{b}{y-scale}}}\\ \mathbf{elif}\;x-scale \leq 7 \cdot 10^{+247}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	9.2
Cost	1616

\[\begin{array}{l} t_0 := \frac{a \cdot b}{x-scale \cdot y-scale} \cdot \frac{-4}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ t_1 := -4 \cdot \frac{\frac{b}{y-scale}}{\frac{\frac{y-scale}{b}}{\frac{a}{x-scale}} \cdot \frac{x-scale}{a}}\\ \mathbf{if}\;x-scale \leq -3.2 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -1.35 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -9.2 \cdot 10^{-128}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{b}{y-scale}}{x-scale \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\\ \mathbf{elif}\;x-scale \leq 1.95 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	9.3
Cost	1616

\[\begin{array}{l} t_0 := a \cdot \frac{b}{y-scale}\\ t_1 := -4 \cdot \frac{\frac{b}{y-scale}}{\frac{\frac{y-scale}{b}}{\frac{a}{x-scale}} \cdot \frac{x-scale}{a}}\\ t_2 := \frac{-4}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \mathbf{if}\;x-scale \leq -3.8 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -3.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{a \cdot b}{x-scale \cdot y-scale} \cdot t_2\\ \mathbf{elif}\;x-scale \leq 8.5 \cdot 10^{-128}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{a}{x-scale \cdot y-scale}}{\frac{x-scale}{t_0}}\\ \mathbf{elif}\;x-scale \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{t_0}{x-scale} \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	8.9
Cost	1484

\[\begin{array}{l} t_0 := \frac{\frac{y-scale}{b}}{\frac{a}{x-scale}}\\ \mathbf{if}\;x-scale \leq -7 \cdot 10^{+156}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale}}{t_0 \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 5.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \frac{-4}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 1.22 \cdot 10^{+144}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{b}{y-scale}}{x-scale \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{\frac{-4}{\frac{x-scale \cdot y-scale}{a \cdot b}}}{x-scale \cdot y-scale}\\ \end{array} \]

Alternative 10
Error	7.7
Cost	1484

\[\begin{array}{l} t_0 := \frac{\frac{y-scale}{b}}{\frac{a}{x-scale}}\\ \mathbf{if}\;x-scale \leq -5 \cdot 10^{+164}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale}}{t_0 \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;x-scale \leq 2.15 \cdot 10^{-205}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(-4 \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)}{x-scale \cdot y-scale}\\ \mathbf{elif}\;x-scale \leq 6.8 \cdot 10^{+146}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{b}{y-scale}}{x-scale \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \frac{\frac{-4}{\frac{x-scale \cdot y-scale}{a \cdot b}}}{x-scale \cdot y-scale}\\ \end{array} \]

Alternative 11
Error	11.8
Cost	1353

\[\begin{array}{l} t_0 := \frac{a}{x-scale \cdot y-scale}\\ \mathbf{if}\;x-scale \leq -2.5 \cdot 10^{+166} \lor \neg \left(x-scale \leq 1.22 \cdot 10^{-178}\right):\\ \;\;\;\;-4 \cdot \left(b \cdot \left(t_0 \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{t_0}{\frac{x-scale}{\frac{a \cdot b}{y-scale}}}\right)\\ \end{array} \]

Alternative 12
Error	10.9
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -5 \cdot 10^{+164} \lor \neg \left(x-scale \leq 1.15 \cdot 10^{-129}\right):\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a}{x-scale \cdot y-scale}}{\frac{x-scale}{\frac{a \cdot b}{y-scale}}}\right)\\ \end{array} \]

Alternative 13
Error	10.4
Cost	1352

\[\begin{array}{l} t_0 := x-scale \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)\\ \mathbf{if}\;x-scale \leq -5.3 \cdot 10^{+163}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{t_0}\right)\\ \mathbf{elif}\;x-scale \leq 2.5 \cdot 10^{-205}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a}{x-scale \cdot y-scale}}{\frac{x-scale}{\frac{a \cdot b}{y-scale}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{b}{y-scale}}{t_0}\\ \end{array} \]

Alternative 14
Error	10.4
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -4 \cdot 10^{+164}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}\right)\\ \mathbf{elif}\;x-scale \leq 5.9 \cdot 10^{-206}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a}{x-scale \cdot y-scale}}{\frac{x-scale}{\frac{a \cdot b}{y-scale}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{b}{y-scale}}{x-scale \cdot \frac{\frac{y-scale}{b}}{\frac{a}{x-scale}}}\\ \end{array} \]

Alternative 15
Error	12.4
Cost	1088

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

Alternative 16
Error	12.5
Cost	1088

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

Alternative 17
Error	30.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023045 
(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))))