Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.0 → 6.6

Time: 1.2min

Precision: binary64

Cost: 7560

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -8.2 \cdot 10^{-160}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \mathbf{elif}\;x-scale \leq -1.1 \cdot 10^{-292}:\\ \;\;\;\;\frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale} \cdot \frac{-4}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}\right)\\ \end{array} \]

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 (<= x-scale -8.2e-160)
   (* -4.0 (/ (* (/ a y-scale) (/ b x-scale)) (/ (/ x-scale b) (/ a y-scale))))
   (if (<= x-scale -1.1e-292)
     (* (/ (pow (* a b) 2.0) (* x-scale y-scale)) (/ -4.0 (* x-scale y-scale)))
     (*
      -4.0
      (* (/ (/ a y-scale) (/ x-scale b)) (/ (/ b x-scale) (/ y-scale a)))))))

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 (x_45_scale <= -8.2e-160) {
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((x_45_scale / b) / (a / y_45_scale)));
	} else if (x_45_scale <= -1.1e-292) {
		tmp = (pow((a * b), 2.0) / (x_45_scale * y_45_scale)) * (-4.0 / (x_45_scale * y_45_scale));
	} else {
		tmp = -4.0 * (((a / y_45_scale) / (x_45_scale / b)) * ((b / x_45_scale) / (y_45_scale / a)));
	}
	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 (x_45_scale <= -8.2e-160) {
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((x_45_scale / b) / (a / y_45_scale)));
	} else if (x_45_scale <= -1.1e-292) {
		tmp = (Math.pow((a * b), 2.0) / (x_45_scale * y_45_scale)) * (-4.0 / (x_45_scale * y_45_scale));
	} else {
		tmp = -4.0 * (((a / y_45_scale) / (x_45_scale / b)) * ((b / x_45_scale) / (y_45_scale / a)));
	}
	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 x_45_scale <= -8.2e-160:
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((x_45_scale / b) / (a / y_45_scale)))
	elif x_45_scale <= -1.1e-292:
		tmp = (math.pow((a * b), 2.0) / (x_45_scale * y_45_scale)) * (-4.0 / (x_45_scale * y_45_scale))
	else:
		tmp = -4.0 * (((a / y_45_scale) / (x_45_scale / b)) * ((b / x_45_scale) / (y_45_scale / a)))
	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 (x_45_scale <= -8.2e-160)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale)) / Float64(Float64(x_45_scale / b) / Float64(a / y_45_scale))));
	elseif (x_45_scale <= -1.1e-292)
		tmp = Float64(Float64((Float64(a * b) ^ 2.0) / Float64(x_45_scale * y_45_scale)) * Float64(-4.0 / Float64(x_45_scale * y_45_scale)));
	else
		tmp = 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))));
	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 (x_45_scale <= -8.2e-160)
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((x_45_scale / b) / (a / y_45_scale)));
	elseif (x_45_scale <= -1.1e-292)
		tmp = (((a * b) ^ 2.0) / (x_45_scale * y_45_scale)) * (-4.0 / (x_45_scale * y_45_scale));
	else
		tmp = -4.0 * (((a / y_45_scale) / (x_45_scale / b)) * ((b / x_45_scale) / (y_45_scale / a)));
	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[x$45$scale, -8.2e-160], N[(-4.0 * N[(N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale / b), $MachinePrecision] / N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -1.1e-292], N[(N[(N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -8.20000000000000003e-160

   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 17.7

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

      [Start] 34.7	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 34.7	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 34.9	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 34.9	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 34.9	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 34.9	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 25.9	\[ \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 25.9	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 25.9	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot -4\right) \]
      times-frac [=>] 17.7	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Taylor expanded in a around 0 34.7

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

   5. Simplified 4.9

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

      [Start] 34.7	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 34.7	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      times-frac [=>] 34.9	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 34.9	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 34.9	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 25.9	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 25.9	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 25.9	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 17.7	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 4.9	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      unpow2 [<=] 4.9	\[ -4 \cdot \color{blue}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}} \]
      associate-*r/ [=>] 4.9	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}}^{2} \]
      associate-/l* [=>] 4.9	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}}^{2} \]

   6. Applied egg-rr 5.0

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

   if -8.20000000000000003e-160 < x-scale < -1.10000000000000006e-292

   1. Initial program 52.2

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

   2. Taylor expanded in angle around 0 63.3

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

   3. Simplified 63.3

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

      [Start] 63.3	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      associate-*r/ [=>] 63.3	\[ \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      unpow2 [=>] 63.3	\[ \frac{-4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 63.3	\[ \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 63.3	\[ \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]
      unpow2 [=>] 63.3	\[ \frac{-4 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]

   4. Applied egg-rr 22.4

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

   if -1.10000000000000006e-292 < x-scale

   1. Initial program 41.4

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

   2. Taylor expanded in angle around 0 39.8

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

   3. Simplified 19.7

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

      [Start] 39.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 39.8	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 40.0	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 40.0	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 40.0	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 40.0	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      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 [=>] 19.7	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Taylor expanded in a around 0 39.8

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

   5. Simplified 5.8

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

      [Start] 39.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 39.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      times-frac [=>] 40.0	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 40.0	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 40.0	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 32.1	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 32.1	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 32.1	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 19.7	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 5.8	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      unpow2 [<=] 5.8	\[ -4 \cdot \color{blue}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}} \]
      associate-*r/ [=>] 6.2	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}}^{2} \]
      associate-/l* [=>] 5.8	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}}^{2} \]

   6. Applied egg-rr 8.2

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

   7. Applied egg-rr 6.0

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -8.2 \cdot 10^{-160}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \mathbf{elif}\;x-scale \leq -1.1 \cdot 10^{-292}:\\ \;\;\;\;\frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale} \cdot \frac{-4}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	26.1
Cost	1485

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -2 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq -3.6 \cdot 10^{-107} \lor \neg \left(x-scale \leq 1.05 \cdot 10^{-214}\right):\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{x-scale}}{x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	20.2
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -3.5 \cdot 10^{-107} \lor \neg \left(x-scale \leq 2.2 \cdot 10^{-261}\right):\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{x-scale}}{x-scale}\right) \cdot \frac{a \cdot \frac{b}{y-scale}}{y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	5.9
Cost	1352

\[\begin{array}{l} t_0 := \frac{x-scale \cdot y-scale}{a \cdot b}\\ \mathbf{if}\;x-scale \leq -1.15 \cdot 10^{-92}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \mathbf{elif}\;x-scale \leq -1.7 \cdot 10^{-294}:\\ \;\;\;\;\frac{-4}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}\right)\\ \end{array} \]

Alternative 4
Error	8.2
Cost	1088

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

Alternative 5
Error	8.2
Cost	1088

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

Alternative 6
Error	5.9
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 7
Error	30.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023066 
(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))))