Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.4 → 6.8

Time: 1.3min

Precision: binary64

Cost: 1616

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y-scale}{b} \cdot \frac{x-scale}{a}\\ t_1 := -4 \cdot \left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}\right)\\ t_2 := -4 \cdot \frac{a}{x-scale}\\ \mathbf{if}\;angle \leq -1.7 \cdot 10^{+169}:\\ \;\;\;\;\frac{t_2 \cdot \frac{b}{y-scale}}{t_0}\\ \mathbf{elif}\;angle \leq -3.2 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;angle \leq 5.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{-4}{\frac{x-scale}{a \cdot \frac{b}{y-scale}}}}{t_0}\\ \mathbf{elif}\;angle \leq 10^{+218}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b}}\\ \end{array} \]

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 (* (/ y-scale b) (/ x-scale a)))
        (t_1
         (*
          -4.0
          (* (/ (/ a y-scale) (/ x-scale b)) (/ (/ b x-scale) (/ y-scale a)))))
        (t_2 (* -4.0 (/ a x-scale))))
   (if (<= angle -1.7e+169)
     (/ (* t_2 (/ b y-scale)) t_0)
     (if (<= angle -3.2e-157)
       t_1
       (if (<= angle 5.1e-86)
         (/ (/ -4.0 (/ x-scale (* a (/ b y-scale)))) t_0)
         (if (<= angle 1e+218)
           t_1
           (* t_2 (/ (* (/ a x-scale) (/ b y-scale)) (/ y-scale b)))))))))

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 = (y_45_scale / b) * (x_45_scale / a);
	double t_1 = -4.0 * (((a / y_45_scale) / (x_45_scale / b)) * ((b / x_45_scale) / (y_45_scale / a)));
	double t_2 = -4.0 * (a / x_45_scale);
	double tmp;
	if (angle <= -1.7e+169) {
		tmp = (t_2 * (b / y_45_scale)) / t_0;
	} else if (angle <= -3.2e-157) {
		tmp = t_1;
	} else if (angle <= 5.1e-86) {
		tmp = (-4.0 / (x_45_scale / (a * (b / y_45_scale)))) / t_0;
	} else if (angle <= 1e+218) {
		tmp = t_1;
	} else {
		tmp = t_2 * (((a / x_45_scale) * (b / y_45_scale)) / (y_45_scale / b));
	}
	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 = (y_45_scale / b) * (x_45_scale / a);
	double t_1 = -4.0 * (((a / y_45_scale) / (x_45_scale / b)) * ((b / x_45_scale) / (y_45_scale / a)));
	double t_2 = -4.0 * (a / x_45_scale);
	double tmp;
	if (angle <= -1.7e+169) {
		tmp = (t_2 * (b / y_45_scale)) / t_0;
	} else if (angle <= -3.2e-157) {
		tmp = t_1;
	} else if (angle <= 5.1e-86) {
		tmp = (-4.0 / (x_45_scale / (a * (b / y_45_scale)))) / t_0;
	} else if (angle <= 1e+218) {
		tmp = t_1;
	} else {
		tmp = t_2 * (((a / x_45_scale) * (b / y_45_scale)) / (y_45_scale / b));
	}
	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 = (y_45_scale / b) * (x_45_scale / a)
	t_1 = -4.0 * (((a / y_45_scale) / (x_45_scale / b)) * ((b / x_45_scale) / (y_45_scale / a)))
	t_2 = -4.0 * (a / x_45_scale)
	tmp = 0
	if angle <= -1.7e+169:
		tmp = (t_2 * (b / y_45_scale)) / t_0
	elif angle <= -3.2e-157:
		tmp = t_1
	elif angle <= 5.1e-86:
		tmp = (-4.0 / (x_45_scale / (a * (b / y_45_scale)))) / t_0
	elif angle <= 1e+218:
		tmp = t_1
	else:
		tmp = t_2 * (((a / x_45_scale) * (b / y_45_scale)) / (y_45_scale / b))
	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(Float64(y_45_scale / b) * Float64(x_45_scale / a))
	t_1 = Float64(-4.0 * Float64(Float64(Float64(a / y_45_scale) / Float64(x_45_scale / b)) * Float64(Float64(b / x_45_scale) / Float64(y_45_scale / a))))
	t_2 = Float64(-4.0 * Float64(a / x_45_scale))
	tmp = 0.0
	if (angle <= -1.7e+169)
		tmp = Float64(Float64(t_2 * Float64(b / y_45_scale)) / t_0);
	elseif (angle <= -3.2e-157)
		tmp = t_1;
	elseif (angle <= 5.1e-86)
		tmp = Float64(Float64(-4.0 / Float64(x_45_scale / Float64(a * Float64(b / y_45_scale)))) / t_0);
	elseif (angle <= 1e+218)
		tmp = t_1;
	else
		tmp = Float64(t_2 * Float64(Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale)) / Float64(y_45_scale / b)));
	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 = (y_45_scale / b) * (x_45_scale / a);
	t_1 = -4.0 * (((a / y_45_scale) / (x_45_scale / b)) * ((b / x_45_scale) / (y_45_scale / a)));
	t_2 = -4.0 * (a / x_45_scale);
	tmp = 0.0;
	if (angle <= -1.7e+169)
		tmp = (t_2 * (b / y_45_scale)) / t_0;
	elseif (angle <= -3.2e-157)
		tmp = t_1;
	elseif (angle <= 5.1e-86)
		tmp = (-4.0 / (x_45_scale / (a * (b / y_45_scale)))) / t_0;
	elseif (angle <= 1e+218)
		tmp = t_1;
	else
		tmp = t_2 * (((a / x_45_scale) * (b / y_45_scale)) / (y_45_scale / b));
	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[(N[(y$45$scale / b), $MachinePrecision] * N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(N[(N[(a / y$45$scale), $MachinePrecision] / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision] * N[(N[(b / x$45$scale), $MachinePrecision] / N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -1.7e+169], N[(N[(t$95$2 * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[angle, -3.2e-157], t$95$1, If[LessEqual[angle, 5.1e-86], N[(N[(-4.0 / N[(x$45$scale / N[(a * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[angle, 1e+218], t$95$1, N[(t$95$2 * N[(N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if angle < -1.70000000000000014e169

   1. Initial program 43.6

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

   2. Simplified 49.0

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

      [Start] 43.6	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 43.6	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 43.6	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 38.4

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

   4. Simplified 17.8

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

      [Start] 38.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 39.0	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      associate-*r* [=>] 39.0	\[ \color{blue}{\left(-4 \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}}} \]
      unpow2 [=>] 39.0	\[ \left(-4 \cdot \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 39.0	\[ \left(-4 \cdot \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      times-frac [=>] 31.3	\[ \left(-4 \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 31.3	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \]
      unpow2 [=>] 31.3	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \]
      times-frac [=>] 17.8	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \]

   5. Applied egg-rr 17.9

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

   6. Simplified 17.8

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

      [Start] 17.9	\[ \left(-4 \cdot \frac{1}{\frac{x-scale}{a} \cdot \frac{x-scale}{a}}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \]
      associate-/r* [=>] 17.8	\[ \left(-4 \cdot \color{blue}{\frac{\frac{1}{\frac{x-scale}{a}}}{\frac{x-scale}{a}}}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \]

   7. Applied egg-rr 4.8

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

   if -1.70000000000000014e169 < angle < -3.20000000000000021e-157 or 5.10000000000000006e-86 < angle < 1.00000000000000008e218

   1. Initial program 43.4

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

   2. Taylor expanded in angle around 0 39.3

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

   3. Simplified 20.5

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

      [Start] 39.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 39.3	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 39.8	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 39.8	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 39.8	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 39.8	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 32.1	\[ \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 32.1	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 32.1	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot -4\right) \]
      times-frac [=>] 20.5	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Taylor expanded in a around 0 39.3

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

   5. Simplified 6.3

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

      [Start] 39.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 39.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      times-frac [=>] 39.8	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 39.8	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 39.8	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 32.0	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 32.0	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 32.0	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 20.5	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 6.3	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      unpow2 [<=] 6.3	\[ -4 \cdot \color{blue}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}} \]
      associate-*r/ [=>] 5.9	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}}^{2} \]
      associate-/l* [=>] 6.3	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}}^{2} \]

   6. Applied egg-rr 9.2

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

   7. Applied egg-rr 6.5

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

   if -3.20000000000000021e-157 < angle < 5.10000000000000006e-86

   1. Initial program 36.2

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

   2. Simplified 42.3

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

      [Start] 36.2	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 36.2	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 36.2	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 40.8

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

   4. Simplified 19.0

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

      [Start] 40.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 40.8	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      associate-*r* [=>] 40.8	\[ \color{blue}{\left(-4 \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}}} \]
      unpow2 [=>] 40.8	\[ \left(-4 \cdot \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 40.8	\[ \left(-4 \cdot \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      times-frac [=>] 32.3	\[ \left(-4 \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 32.3	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \]
      unpow2 [=>] 32.3	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \]
      times-frac [=>] 19.0	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \]

   5. Applied egg-rr 19.2

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

   6. Simplified 19.0

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

      [Start] 19.2	\[ \left(-4 \cdot \frac{1}{\frac{x-scale}{a} \cdot \frac{x-scale}{a}}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \]
      associate-/r* [=>] 19.0	\[ \left(-4 \cdot \color{blue}{\frac{\frac{1}{\frac{x-scale}{a}}}{\frac{x-scale}{a}}}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \]

   7. Applied egg-rr 6.1

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

   8. Applied egg-rr 7.5

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

   if 1.00000000000000008e218 < angle

   1. Initial program 44.9

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

   2. Simplified 47.9

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

      [Start] 44.9	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 44.9	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 44.9	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 39.0

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

   4. Simplified 21.2

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

      [Start] 39.0	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 40.2	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      associate-*r* [=>] 40.2	\[ \color{blue}{\left(-4 \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}}} \]
      unpow2 [=>] 40.2	\[ \left(-4 \cdot \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 40.2	\[ \left(-4 \cdot \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      times-frac [=>] 33.0	\[ \left(-4 \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 33.0	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \]
      unpow2 [=>] 33.0	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \]
      times-frac [=>] 21.2	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \]

   5. Applied egg-rr 21.4

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

   6. Simplified 21.2

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

      [Start] 21.4	\[ \left(-4 \cdot \frac{1}{\frac{x-scale}{a} \cdot \frac{x-scale}{a}}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \]
      associate-/r* [=>] 21.2	\[ \left(-4 \cdot \color{blue}{\frac{\frac{1}{\frac{x-scale}{a}}}{\frac{x-scale}{a}}}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \]

   7. Applied egg-rr 6.5

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

   8. Applied egg-rr 9.0

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -1.7 \cdot 10^{+169}:\\ \;\;\;\;\frac{\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;angle \leq -3.2 \cdot 10^{-157}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}\right)\\ \mathbf{elif}\;angle \leq 5.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{-4}{\frac{x-scale}{a \cdot \frac{b}{y-scale}}}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;angle \leq 10^{+218}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.4
Cost	7820

\[\begin{array}{l} t_0 := \frac{y-scale}{b} \cdot \frac{x-scale}{a}\\ t_1 := \frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+168}:\\ \;\;\;\;\frac{\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{t_0}\\ \mathbf{elif}\;\frac{angle}{180} \leq -4 \cdot 10^{-152}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-50}:\\ \;\;\;\;\frac{\frac{-4}{\frac{x-scale}{a \cdot \frac{b}{y-scale}}}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {t_1}^{2}\\ \end{array} \]

Alternative 2
Error	6.6
Cost	1616

\[\begin{array}{l} t_0 := -4 \cdot \left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}\right)\\ t_1 := \left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b}}\\ \mathbf{if}\;a \leq -4 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-240}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	6.4
Cost	1616

\[\begin{array}{l} t_0 := -4 \cdot \frac{a}{x-scale}\\ t_1 := \frac{t_0 \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ t_2 := -4 \cdot \left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}\right)\\ \mathbf{if}\;angle \leq -2.1 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;angle \leq -1.1 \cdot 10^{-156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;angle \leq 6.1 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;angle \leq 4.4 \cdot 10^{+218}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b}}\\ \end{array} \]

Alternative 4
Error	7.0
Cost	1485

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.4 \cdot 10^{+125} \lor \neg \left(angle \leq -5.8 \cdot 10^{-153}\right) \land angle \leq 6.1 \cdot 10^{-209}:\\ \;\;\;\;\frac{-4}{\frac{y-scale}{b}} \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \end{array} \]

Alternative 5
Error	12.5
Cost	1088

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

Alternative 6
Error	6.0
Cost	1088

\[\begin{array}{l} t_0 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ -4 \cdot \left(t_0 \cdot t_0\right) \end{array} \]

Alternative 7
Error	6.1
Cost	1088

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

Alternative 8
Error	30.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023073 
(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))))