Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.4 → 5.9

Time: 1.2min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := -4 \cdot {\left(\frac{b}{\frac{x-scale}{a} \cdot y-scale}\right)}^{2}\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{+256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-197}:\\ \;\;\;\;-4 \cdot {\left(\frac{a \cdot b}{x-scale \cdot y-scale}\right)}^{2}\\ \mathbf{elif}\;a \leq -2.05 \cdot 10^{-272}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{\frac{x-scale}{\frac{a}{y-scale}}}\right)}^{2}\\ \end{array} \]

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
 (let* ((t_0 (* -4.0 (pow (/ b (* (/ x-scale a) y-scale)) 2.0))))
   (if (<= a -1.8e+256)
     t_0
     (if (<= a -2.8e-197)
       (* -4.0 (pow (/ (* a b) (* x-scale y-scale)) 2.0))
       (if (<= a -2.05e-272)
         t_0
         (if (<= a 3.8e-97)
           (* -4.0 (pow (/ a (* x-scale (/ y-scale b))) 2.0))
           (* -4.0 (pow (/ b (/ x-scale (/ a y-scale))) 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 t_0 = -4.0 * pow((b / ((x_45_scale / a) * y_45_scale)), 2.0);
	double tmp;
	if (a <= -1.8e+256) {
		tmp = t_0;
	} else if (a <= -2.8e-197) {
		tmp = -4.0 * pow(((a * b) / (x_45_scale * y_45_scale)), 2.0);
	} else if (a <= -2.05e-272) {
		tmp = t_0;
	} else if (a <= 3.8e-97) {
		tmp = -4.0 * pow((a / (x_45_scale * (y_45_scale / b))), 2.0);
	} else {
		tmp = -4.0 * pow((b / (x_45_scale / (a / y_45_scale))), 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 t_0 = -4.0 * Math.pow((b / ((x_45_scale / a) * y_45_scale)), 2.0);
	double tmp;
	if (a <= -1.8e+256) {
		tmp = t_0;
	} else if (a <= -2.8e-197) {
		tmp = -4.0 * Math.pow(((a * b) / (x_45_scale * y_45_scale)), 2.0);
	} else if (a <= -2.05e-272) {
		tmp = t_0;
	} else if (a <= 3.8e-97) {
		tmp = -4.0 * Math.pow((a / (x_45_scale * (y_45_scale / b))), 2.0);
	} else {
		tmp = -4.0 * Math.pow((b / (x_45_scale / (a / y_45_scale))), 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):
	t_0 = -4.0 * math.pow((b / ((x_45_scale / a) * y_45_scale)), 2.0)
	tmp = 0
	if a <= -1.8e+256:
		tmp = t_0
	elif a <= -2.8e-197:
		tmp = -4.0 * math.pow(((a * b) / (x_45_scale * y_45_scale)), 2.0)
	elif a <= -2.05e-272:
		tmp = t_0
	elif a <= 3.8e-97:
		tmp = -4.0 * math.pow((a / (x_45_scale * (y_45_scale / b))), 2.0)
	else:
		tmp = -4.0 * math.pow((b / (x_45_scale / (a / y_45_scale))), 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)
	t_0 = Float64(-4.0 * (Float64(b / Float64(Float64(x_45_scale / a) * y_45_scale)) ^ 2.0))
	tmp = 0.0
	if (a <= -1.8e+256)
		tmp = t_0;
	elseif (a <= -2.8e-197)
		tmp = Float64(-4.0 * (Float64(Float64(a * b) / Float64(x_45_scale * y_45_scale)) ^ 2.0));
	elseif (a <= -2.05e-272)
		tmp = t_0;
	elseif (a <= 3.8e-97)
		tmp = Float64(-4.0 * (Float64(a / Float64(x_45_scale * Float64(y_45_scale / b))) ^ 2.0));
	else
		tmp = Float64(-4.0 * (Float64(b / Float64(x_45_scale / Float64(a / y_45_scale))) ^ 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)
	t_0 = -4.0 * ((b / ((x_45_scale / a) * y_45_scale)) ^ 2.0);
	tmp = 0.0;
	if (a <= -1.8e+256)
		tmp = t_0;
	elseif (a <= -2.8e-197)
		tmp = -4.0 * (((a * b) / (x_45_scale * y_45_scale)) ^ 2.0);
	elseif (a <= -2.05e-272)
		tmp = t_0;
	elseif (a <= 3.8e-97)
		tmp = -4.0 * ((a / (x_45_scale * (y_45_scale / b))) ^ 2.0);
	else
		tmp = -4.0 * ((b / (x_45_scale / (a / y_45_scale))) ^ 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_] := Block[{t$95$0 = N[(-4.0 * N[Power[N[(b / N[(N[(x$45$scale / a), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.8e+256], t$95$0, If[LessEqual[a, -2.8e-197], N[(-4.0 * N[Power[N[(N[(a * b), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.05e-272], t$95$0, If[LessEqual[a, 3.8e-97], N[(-4.0 * N[Power[N[(a / N[(x$45$scale * N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(b / N[(x$45$scale / N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]), $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 a < -1.79999999999999985e256 or -2.8000000000000002e-197 < a < -2.0499999999999999e-272

   1. Initial program 41.1

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

   2. Taylor expanded in angle around 0 43.8

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

   3. Simplified 37.9

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

   4. Applied egg-rr 7.9

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

   5. Applied egg-rr 7.0

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

   6. Applied egg-rr 7.3

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

   if -1.79999999999999985e256 < a < -2.8000000000000002e-197

   1. Initial program 41.3

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

   2. Taylor expanded in angle around 0 37.6

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

   3. Simplified 28.9

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

   4. Applied egg-rr 4.8

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

   5. Applied egg-rr 5.3

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

   6. Taylor expanded in b around 0 5.0

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

   if -2.0499999999999999e-272 < a < 3.8000000000000001e-97

   1. Initial program 34.6

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

   2. Taylor expanded in angle around 0 37.3

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

   3. Simplified 28.9

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

   4. Applied egg-rr 5.6

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

   5. Applied egg-rr 5.0

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

   6. Taylor expanded in b around 0 5.0

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

   7. Simplified 6.1

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

   if 3.8000000000000001e-97 < a

   1. Initial program 48.3

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

   2. Taylor expanded in angle around 0 43.6

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

   3. Simplified 35.2

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

   4. Applied egg-rr 6.4

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

   5. Applied egg-rr 6.1

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{+256}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{\frac{x-scale}{a} \cdot y-scale}\right)}^{2}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-197}:\\ \;\;\;\;-4 \cdot {\left(\frac{a \cdot b}{x-scale \cdot y-scale}\right)}^{2}\\ \mathbf{elif}\;a \leq -2.05 \cdot 10^{-272}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{\frac{x-scale}{a} \cdot y-scale}\right)}^{2}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-97}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{\frac{x-scale}{\frac{a}{y-scale}}}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(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))))