Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.3 → 5.9

Time: 1.1min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-200}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-161}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)}^{2}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+89}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot \frac{a}{x-scale}}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left({\left(x-scale \cdot \frac{y-scale}{b \cdot a}\right)}^{-1}\right)}^{2}\\ \end{array} \]

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
 (if (<= b -1.65e-200)
   (* -4.0 (pow (* (/ a x-scale) (/ b y-scale)) 2.0))
   (if (<= b 3.4e-161)
     (* -4.0 (pow (/ (* b a) (* x-scale y-scale)) 2.0))
     (if (<= b 9.5e+89)
       (* -4.0 (pow (/ (* b (/ a x-scale)) y-scale) 2.0))
       (* -4.0 (pow (pow (* x-scale (/ y-scale (* b a))) -1.0) 2.0))))))

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 tmp;
	if (b <= -1.65e-200) {
		tmp = -4.0 * pow(((a / x_45_scale) * (b / y_45_scale)), 2.0);
	} else if (b <= 3.4e-161) {
		tmp = -4.0 * pow(((b * a) / (x_45_scale * y_45_scale)), 2.0);
	} else if (b <= 9.5e+89) {
		tmp = -4.0 * pow(((b * (a / x_45_scale)) / y_45_scale), 2.0);
	} else {
		tmp = -4.0 * pow(pow((x_45_scale * (y_45_scale / (b * a))), -1.0), 2.0);
	}
	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 tmp;
	if (b <= -1.65e-200) {
		tmp = -4.0 * Math.pow(((a / x_45_scale) * (b / y_45_scale)), 2.0);
	} else if (b <= 3.4e-161) {
		tmp = -4.0 * Math.pow(((b * a) / (x_45_scale * y_45_scale)), 2.0);
	} else if (b <= 9.5e+89) {
		tmp = -4.0 * Math.pow(((b * (a / x_45_scale)) / y_45_scale), 2.0);
	} else {
		tmp = -4.0 * Math.pow(Math.pow((x_45_scale * (y_45_scale / (b * a))), -1.0), 2.0);
	}
	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):
	tmp = 0
	if b <= -1.65e-200:
		tmp = -4.0 * math.pow(((a / x_45_scale) * (b / y_45_scale)), 2.0)
	elif b <= 3.4e-161:
		tmp = -4.0 * math.pow(((b * a) / (x_45_scale * y_45_scale)), 2.0)
	elif b <= 9.5e+89:
		tmp = -4.0 * math.pow(((b * (a / x_45_scale)) / y_45_scale), 2.0)
	else:
		tmp = -4.0 * math.pow(math.pow((x_45_scale * (y_45_scale / (b * a))), -1.0), 2.0)
	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)
	tmp = 0.0
	if (b <= -1.65e-200)
		tmp = Float64(-4.0 * (Float64(Float64(a / x_45_scale) * Float64(b / y_45_scale)) ^ 2.0));
	elseif (b <= 3.4e-161)
		tmp = Float64(-4.0 * (Float64(Float64(b * a) / Float64(x_45_scale * y_45_scale)) ^ 2.0));
	elseif (b <= 9.5e+89)
		tmp = Float64(-4.0 * (Float64(Float64(b * Float64(a / x_45_scale)) / y_45_scale) ^ 2.0));
	else
		tmp = Float64(-4.0 * ((Float64(x_45_scale * Float64(y_45_scale / Float64(b * a))) ^ -1.0) ^ 2.0));
	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)
	tmp = 0.0;
	if (b <= -1.65e-200)
		tmp = -4.0 * (((a / x_45_scale) * (b / y_45_scale)) ^ 2.0);
	elseif (b <= 3.4e-161)
		tmp = -4.0 * (((b * a) / (x_45_scale * y_45_scale)) ^ 2.0);
	elseif (b <= 9.5e+89)
		tmp = -4.0 * (((b * (a / x_45_scale)) / y_45_scale) ^ 2.0);
	else
		tmp = -4.0 * (((x_45_scale * (y_45_scale / (b * a))) ^ -1.0) ^ 2.0);
	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_] := If[LessEqual[b, -1.65e-200], N[(-4.0 * N[Power[N[(N[(a / x$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.4e-161], N[(-4.0 * N[Power[N[(N[(b * a), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.5e+89], N[(-4.0 * N[Power[N[(N[(b * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[Power[N[(x$45$scale * N[(y$45$scale / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Bits error versus x-scale

Bits error versus y-scale

Derivation

1. Split input into 4 regimes

2. if b < -1.6499999999999999e-200

   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. Taylor expanded in angle around 0 41.0

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

   3. Simplified 33.4

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

   4. Applied egg-rr 6.0

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

   5. Taylor expanded in a around 0 5.7

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

   6. Simplified 6.4

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

   if -1.6499999999999999e-200 < b < 3.39999999999999982e-161

   1. Initial program 33.4

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

   2. Taylor expanded in angle around 0 36.7

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

   3. Simplified 26.6

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

   4. Applied egg-rr 5.8

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

   5. Applied egg-rr 6.4

      \[\leadsto \color{blue}{{\left({\left(\frac{b \cdot \frac{a}{x-scale}}{y-scale}\right)}^{2} \cdot -4\right)}^{1}} \]

   6. Taylor expanded in b around inf 5.4

      \[\leadsto {\left({\color{blue}{\left(\frac{a \cdot b}{x-scale \cdot y-scale}\right)}}^{2} \cdot -4\right)}^{1} \]

   if 3.39999999999999982e-161 < b < 9.5000000000000003e89

   1. Initial program 38.8

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

   2. Taylor expanded in angle around 0 34.7

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

   3. Simplified 31.9

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

   4. Applied egg-rr 3.5

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

   5. Applied egg-rr 3.6

      \[\leadsto \color{blue}{{\left({\left(\frac{b \cdot \frac{a}{x-scale}}{y-scale}\right)}^{2} \cdot -4\right)}^{1}} \]

   if 9.5000000000000003e89 < b

   1. Initial program 58.5

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

   2. Taylor expanded in angle around 0 54.4

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

   3. Simplified 37.2

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

   4. Applied egg-rr 10.3

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

   5. Applied egg-rr 9.1

      \[\leadsto \color{blue}{{\left({\left(\frac{b \cdot \frac{a}{x-scale}}{y-scale}\right)}^{2} \cdot -4\right)}^{1}} \]

   6. Applied egg-rr 9.7

      \[\leadsto {\left({\color{blue}{\left({\left(\frac{y-scale}{b \cdot a} \cdot x-scale\right)}^{-1}\right)}}^{2} \cdot -4\right)}^{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-200}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-161}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)}^{2}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+89}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot \frac{a}{x-scale}}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left({\left(x-scale \cdot \frac{y-scale}{b \cdot a}\right)}^{-1}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022165 
(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))))