Simplification of discriminant from scale-rotated-ellipse

Average Error: 40.7 → 5.7

Time: 1.3min

Precision: binary64

Cost: 7172

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 9.2 \cdot 10^{-199}:\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \end{array} \]

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
 (if (<= y-scale 9.2e-199)
   (* -4.0 (pow (/ (* (/ a y-scale) b) x-scale) 2.0))
   (* -4.0 (pow (* (/ b y-scale) (/ a x-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 tmp;
	if (y_45_scale <= 9.2e-199) {
		tmp = -4.0 * pow((((a / y_45_scale) * b) / x_45_scale), 2.0);
	} else {
		tmp = -4.0 * pow(((b / y_45_scale) * (a / x_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 tmp;
	if (y_45_scale <= 9.2e-199) {
		tmp = -4.0 * Math.pow((((a / y_45_scale) * b) / x_45_scale), 2.0);
	} else {
		tmp = -4.0 * Math.pow(((b / y_45_scale) * (a / x_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):
	tmp = 0
	if y_45_scale <= 9.2e-199:
		tmp = -4.0 * math.pow((((a / y_45_scale) * b) / x_45_scale), 2.0)
	else:
		tmp = -4.0 * math.pow(((b / y_45_scale) * (a / x_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)
	tmp = 0.0
	if (y_45_scale <= 9.2e-199)
		tmp = Float64(-4.0 * (Float64(Float64(Float64(a / y_45_scale) * b) / x_45_scale) ^ 2.0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(b / y_45_scale) * Float64(a / x_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)
	tmp = 0.0;
	if (y_45_scale <= 9.2e-199)
		tmp = -4.0 * ((((a / y_45_scale) * b) / x_45_scale) ^ 2.0);
	else
		tmp = -4.0 * (((b / y_45_scale) * (a / x_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_] := If[LessEqual[y$45$scale, 9.2e-199], N[(-4.0 * N[Power[N[(N[(N[(a / y$45$scale), $MachinePrecision] * b), $MachinePrecision] / x$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(b / y$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 9.2000000000000005e-199

   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 41.5

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

   3. Simplified 32.2

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

      [Start] 41.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      associate-/l* [=>] 41.3	\[ -4 \cdot \color{blue}{\frac{{a}^{2}}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{b}^{2}}}} \]
      unpow2 [=>] 41.3	\[ -4 \cdot \frac{\color{blue}{a \cdot a}}{\frac{{y-scale}^{2} \cdot {x-scale}^{2}}{{b}^{2}}} \]
      unpow2 [=>] 41.3	\[ -4 \cdot \frac{a \cdot a}{\frac{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}}{{b}^{2}}} \]
      unpow2 [=>] 41.3	\[ -4 \cdot \frac{a \cdot a}{\frac{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}}{{b}^{2}}} \]
      unswap-sqr [=>] 32.2	\[ -4 \cdot \frac{a \cdot a}{\frac{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}}{{b}^{2}}} \]
      unpow2 [=>] 32.2	\[ -4 \cdot \frac{a \cdot a}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{\color{blue}{b \cdot b}}} \]

   4. Taylor expanded in a around 0 41.5

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

   5. Simplified 6.1

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

      [Start] 41.5	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 41.3	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 41.3	\[ -4 \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 41.3	\[ -4 \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      unpow2 [=>] 41.3	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right) \]
      unpow2 [=>] 41.3	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right) \]
      times-frac [=>] 32.5	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right) \]
      times-frac [=>] 20.0	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 6.2	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      associate-/r/ [<=] 7.9	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \color{blue}{\frac{a}{\frac{y-scale}{\frac{b}{x-scale}}}}\right) \]
      associate-/r/ [<=] 6.4	\[ -4 \cdot \left(\color{blue}{\frac{a}{\frac{y-scale}{\frac{b}{x-scale}}}} \cdot \frac{a}{\frac{y-scale}{\frac{b}{x-scale}}}\right) \]
      unpow2 [<=] 6.4	\[ -4 \cdot \color{blue}{{\left(\frac{a}{\frac{y-scale}{\frac{b}{x-scale}}}\right)}^{2}} \]
      associate-/r/ [=>] 6.2	\[ -4 \cdot {\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}}^{2} \]
      associate-*r/ [=>] 6.1	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}}^{2} \]

   if 9.2000000000000005e-199 < y-scale

   1. Initial program 40.2

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

   2. Simplified 43.5

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

      [Start] 40.2

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

   3. Taylor expanded in angle around 0 37.1

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

   4. Simplified 26.8

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

      [Start] 37.1	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 37.1	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 37.3	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 37.3	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 37.3	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 31.4	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 31.4	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 26.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 26.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Taylor expanded in b around 0 37.1

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

   6. Simplified 5.2

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

      [Start] 37.1	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 37.3	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 37.3	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 37.3	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      associate-*l/ [<=] 33.6	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale \cdot x-scale} \cdot a\right)} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 33.6	\[ -4 \cdot \left(\left(\frac{a}{x-scale \cdot x-scale} \cdot a\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 33.6	\[ -4 \cdot \left(\left(\frac{a}{x-scale \cdot x-scale} \cdot a\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      *-commutative [=>] 33.6	\[ -4 \cdot \left(\color{blue}{\left(a \cdot \frac{a}{x-scale \cdot x-scale}\right)} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      associate-/r* [=>] 29.4	\[ -4 \cdot \left(\left(a \cdot \color{blue}{\frac{\frac{a}{x-scale}}{x-scale}}\right) \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      associate-*r/ [=>] 29.4	\[ -4 \cdot \left(\color{blue}{\frac{a \cdot \frac{a}{x-scale}}{x-scale}} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      associate-*l/ [<=] 27.9	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      unpow2 [<=] 27.9	\[ -4 \cdot \left(\color{blue}{{\left(\frac{a}{x-scale}\right)}^{2}} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      associate-*r/ [=>] 27.7	\[ -4 \cdot \color{blue}{\frac{{\left(\frac{a}{x-scale}\right)}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot y-scale}} \]
      times-frac [=>] 24.2	\[ -4 \cdot \color{blue}{\left(\frac{{\left(\frac{a}{x-scale}\right)}^{2}}{y-scale} \cdot \frac{b \cdot b}{y-scale}\right)} \]
      associate-*l/ [<=] 20.2	\[ -4 \cdot \left(\frac{{\left(\frac{a}{x-scale}\right)}^{2}}{y-scale} \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot b\right)}\right) \]
      associate-/r/ [<=] 20.2	\[ -4 \cdot \left(\frac{{\left(\frac{a}{x-scale}\right)}^{2}}{y-scale} \cdot \color{blue}{\frac{b}{\frac{y-scale}{b}}}\right) \]
      associate-*l/ [=>] 20.0	\[ -4 \cdot \color{blue}{\frac{{\left(\frac{a}{x-scale}\right)}^{2} \cdot \frac{b}{\frac{y-scale}{b}}}{y-scale}} \]
      associate-*r/ [<=] 20.7	\[ -4 \cdot \color{blue}{\left({\left(\frac{a}{x-scale}\right)}^{2} \cdot \frac{\frac{b}{\frac{y-scale}{b}}}{y-scale}\right)} \]
      *-commutative [=>] 20.7	\[ -4 \cdot \color{blue}{\left(\frac{\frac{b}{\frac{y-scale}{b}}}{y-scale} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right)} \]
      associate-/r/ [=>] 20.7	\[ -4 \cdot \left(\frac{\color{blue}{\frac{b}{y-scale} \cdot b}}{y-scale} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right) \]
      associate-*r/ [<=] 18.7	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right) \]
      unpow2 [=>] 18.7	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 9.2 \cdot 10^{-199}:\\ \;\;\;\;-4 \cdot {\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.8
Cost	7172

\[\begin{array}{l} t_0 := \frac{a}{\frac{y-scale}{\frac{b}{x-scale}}}\\ \mathbf{if}\;x-scale \leq -1.8 \cdot 10^{-251}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \end{array} \]

Alternative 2
Error	6.9
Cost	1353

\[\begin{array}{l} t_0 := \frac{a}{\frac{y-scale}{\frac{b}{x-scale}}}\\ \mathbf{if}\;x-scale \leq -5 \cdot 10^{-262} \lor \neg \left(x-scale \leq 8.8 \cdot 10^{-169}\right):\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{a}{x-scale}}{\frac{y-scale}{b} \cdot \left(x-scale \cdot \frac{y-scale}{b}\right)}\\ \end{array} \]

Alternative 3
Error	6.5
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x-scale \leq 1.8 \cdot 10^{-292} \lor \neg \left(x-scale \leq 4.6 \cdot 10^{-163}\right):\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \frac{a}{x-scale}}{\frac{y-scale}{b} \cdot \left(x-scale \cdot \frac{y-scale}{b}\right)}\\ \end{array} \]

Alternative 4
Error	7.0
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x-scale \leq 3.4 \cdot 10^{-292} \lor \neg \left(x-scale \leq 2.7 \cdot 10^{-100}\right):\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{b}{y-scale} \cdot \left(a \cdot \frac{a}{x-scale}\right)}{x-scale \cdot \frac{y-scale}{b}}\\ \end{array} \]

Alternative 5
Error	25.0
Cost	1352

\[\begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+130}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{+142}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a \cdot a}{x-scale}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	7.3
Cost	1352

\[\begin{array}{l} t_0 := \frac{a}{\frac{y-scale}{\frac{b}{x-scale}}}\\ \mathbf{if}\;x-scale \leq 1.65 \cdot 10^{-306}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;x-scale \leq 7.2 \cdot 10^{-92}:\\ \;\;\;\;-4 \cdot \frac{\left(\frac{a}{y-scale} \cdot b\right) \cdot \frac{b}{\frac{y-scale}{\frac{a}{x-scale}}}}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \end{array} \]

Alternative 7
Error	6.9
Cost	1352

\[\begin{array}{l} \mathbf{if}\;angle \leq 5.5 \cdot 10^{-58}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\\ \mathbf{elif}\;angle \leq 6 \cdot 10^{+178}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{\frac{y-scale}{b} \cdot \left(\frac{y-scale}{b} \cdot \frac{x-scale}{a}\right)}}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{\frac{y-scale}{\frac{b}{x-scale}}} \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right)\\ \end{array} \]

Alternative 8
Error	8.9
Cost	1088

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

Alternative 9
Error	5.7
Cost	1088

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

Alternative 10
Error	30.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023018 
(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))))