Simplification of discriminant from scale-rotated-ellipse

Average Error: 63.85% → 9.02%

Time: 1.2min

Precision: binary64

Cost: 1616

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ t_1 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+106}:\\ \;\;\;\;-4 \cdot \frac{t_1}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 10^{-216}:\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \frac{-4 \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \end{array} \]

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
 (let* ((t_0
         (/
          (* (* -4.0 (/ a x-scale)) (/ b y-scale))
          (* (/ y-scale b) (/ x-scale a))))
        (t_1 (* (/ a y-scale) (/ b x-scale))))
   (if (<= b -1.5e+106)
     (* -4.0 (/ t_1 (* (/ y-scale a) (/ x-scale b))))
     (if (<= b -3e-248)
       t_0
       (if (<= b 1e-216)
         (/
          (* (* b a) (/ (* -4.0 (* b a)) (* y-scale x-scale)))
          (* y-scale x-scale))
         (if (<= b 3.8e-50) t_0 (* -4.0 (* t_1 t_1))))))))

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 * (a / x_45_scale)) * (b / y_45_scale)) / ((y_45_scale / b) * (x_45_scale / a));
	double t_1 = (a / y_45_scale) * (b / x_45_scale);
	double tmp;
	if (b <= -1.5e+106) {
		tmp = -4.0 * (t_1 / ((y_45_scale / a) * (x_45_scale / b)));
	} else if (b <= -3e-248) {
		tmp = t_0;
	} else if (b <= 1e-216) {
		tmp = ((b * a) * ((-4.0 * (b * a)) / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale);
	} else if (b <= 3.8e-50) {
		tmp = t_0;
	} else {
		tmp = -4.0 * (t_1 * t_1);
	}
	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 * (a / x_45_scale)) * (b / y_45_scale)) / ((y_45_scale / b) * (x_45_scale / a));
	double t_1 = (a / y_45_scale) * (b / x_45_scale);
	double tmp;
	if (b <= -1.5e+106) {
		tmp = -4.0 * (t_1 / ((y_45_scale / a) * (x_45_scale / b)));
	} else if (b <= -3e-248) {
		tmp = t_0;
	} else if (b <= 1e-216) {
		tmp = ((b * a) * ((-4.0 * (b * a)) / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale);
	} else if (b <= 3.8e-50) {
		tmp = t_0;
	} else {
		tmp = -4.0 * (t_1 * t_1);
	}
	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 * (a / x_45_scale)) * (b / y_45_scale)) / ((y_45_scale / b) * (x_45_scale / a))
	t_1 = (a / y_45_scale) * (b / x_45_scale)
	tmp = 0
	if b <= -1.5e+106:
		tmp = -4.0 * (t_1 / ((y_45_scale / a) * (x_45_scale / b)))
	elif b <= -3e-248:
		tmp = t_0
	elif b <= 1e-216:
		tmp = ((b * a) * ((-4.0 * (b * a)) / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale)
	elif b <= 3.8e-50:
		tmp = t_0
	else:
		tmp = -4.0 * (t_1 * t_1)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(Float64(angle / 180.0) * pi))) * cos(Float64(Float64(angle / 180.0) * pi))) / x_45_scale) / y_45_scale) * Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(Float64(angle / 180.0) * pi))) * cos(Float64(Float64(angle / 180.0) * pi))) / x_45_scale) / y_45_scale)) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(-4.0 * Float64(a / x_45_scale)) * Float64(b / y_45_scale)) / Float64(Float64(y_45_scale / b) * Float64(x_45_scale / a)))
	t_1 = Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale))
	tmp = 0.0
	if (b <= -1.5e+106)
		tmp = Float64(-4.0 * Float64(t_1 / Float64(Float64(y_45_scale / a) * Float64(x_45_scale / b))));
	elseif (b <= -3e-248)
		tmp = t_0;
	elseif (b <= 1e-216)
		tmp = Float64(Float64(Float64(b * a) * Float64(Float64(-4.0 * Float64(b * a)) / Float64(y_45_scale * x_45_scale))) / Float64(y_45_scale * x_45_scale));
	elseif (b <= 3.8e-50)
		tmp = t_0;
	else
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	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 * (a / x_45_scale)) * (b / y_45_scale)) / ((y_45_scale / b) * (x_45_scale / a));
	t_1 = (a / y_45_scale) * (b / x_45_scale);
	tmp = 0.0;
	if (b <= -1.5e+106)
		tmp = -4.0 * (t_1 / ((y_45_scale / a) * (x_45_scale / b)));
	elseif (b <= -3e-248)
		tmp = t_0;
	elseif (b <= 1e-216)
		tmp = ((b * a) * ((-4.0 * (b * a)) / (y_45_scale * x_45_scale))) / (y_45_scale * x_45_scale);
	elseif (b <= 3.8e-50)
		tmp = t_0;
	else
		tmp = -4.0 * (t_1 * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(-4.0 * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale / b), $MachinePrecision] * N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.5e+106], N[(-4.0 * N[(t$95$1 / N[(N[(y$45$scale / a), $MachinePrecision] * N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3e-248], t$95$0, If[LessEqual[b, 1e-216], N[(N[(N[(b * a), $MachinePrecision] * N[(N[(-4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-50], t$95$0, N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.5e106

   1. Initial program 94.97

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

   2. Taylor expanded in angle around 0 88.64

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

   3. Taylor expanded in x-scale around 0 88.64

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

   4. Simplified 13.41

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

      [Start] 88.64	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 89.81	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 89.81	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 89.81	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 89.81	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 89.81	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 87.05	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      times-frac [=>] 45.4	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]
      unswap-sqr [=>] 16.79	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      times-frac [<=] 32.04	\[ -4 \cdot \left(\color{blue}{\frac{a \cdot b}{x-scale \cdot y-scale}} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right) \]
      *-commutative [<=] 32.04	\[ -4 \cdot \left(\frac{a \cdot b}{\color{blue}{y-scale \cdot x-scale}} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right) \]
      times-frac [<=] 16.63	\[ -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{a \cdot b}{x-scale \cdot y-scale}}\right) \]
      *-commutative [<=] 16.63	\[ -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \frac{a \cdot b}{\color{blue}{y-scale \cdot x-scale}}\right) \]
      unpow2 [<=] 16.63	\[ -4 \cdot \color{blue}{{\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2}} \]
      times-frac [=>] 13.41	\[ -4 \cdot {\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}}^{2} \]

   5. Applied egg-rr 13.41

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

   6. Applied egg-rr 13.57

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

   if -1.5e106 < b < -3.00000000000000014e-248 or 1e-216 < b < 3.7999999999999999e-50

   1. Initial program 54.74

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

   2. Simplified 63.19

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

      [Start] 54.74	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 54.74	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 54.74	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 52.98

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

   4. Simplified 26.12

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

      [Start] 52.98	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 52.63	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      associate-*r* [=>] 52.63	\[ \color{blue}{\left(-4 \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}}} \]
      unpow2 [=>] 52.63	\[ \left(-4 \cdot \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 52.63	\[ \left(-4 \cdot \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      times-frac [=>] 38.48	\[ \left(-4 \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \cdot \frac{{b}^{2}}{{y-scale}^{2}} \]
      unpow2 [=>] 38.48	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \]
      unpow2 [=>] 38.48	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \]
      times-frac [=>] 26.12	\[ \left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \]

   5. Applied egg-rr 26.1

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

   6. Applied egg-rr 7.04

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

   if -3.00000000000000014e-248 < b < 1e-216

   1. Initial program 51.2

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

   2. Taylor expanded in angle around 0 54.59

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

   3. Taylor expanded in x-scale around 0 54.59

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

   4. Simplified 8.38

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

      [Start] 54.59	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 54.9	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 54.9	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 54.9	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 54.9	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 54.9	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 42.13	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      times-frac [=>] 24.09	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]
      unswap-sqr [=>] 8.37	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      times-frac [<=] 15.33	\[ -4 \cdot \left(\color{blue}{\frac{a \cdot b}{x-scale \cdot y-scale}} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right) \]
      *-commutative [<=] 15.33	\[ -4 \cdot \left(\frac{a \cdot b}{\color{blue}{y-scale \cdot x-scale}} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right) \]
      times-frac [<=] 7.65	\[ -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{a \cdot b}{x-scale \cdot y-scale}}\right) \]
      *-commutative [<=] 7.65	\[ -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \frac{a \cdot b}{\color{blue}{y-scale \cdot x-scale}}\right) \]
      unpow2 [<=] 7.65	\[ -4 \cdot \color{blue}{{\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2}} \]
      times-frac [=>] 8.38	\[ -4 \cdot {\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}}^{2} \]

   5. Applied egg-rr 8.38

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

   6. Applied egg-rr 10.01

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

   if 3.7999999999999999e-50 < b

   1. Initial program 77.63

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

   2. Taylor expanded in angle around 0 68.85

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

   3. Taylor expanded in x-scale around 0 68.85

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

   4. Simplified 10.4

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

      [Start] 68.85	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 69.67	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 69.66	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 69.66	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 69.66	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 69.66	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      times-frac [=>] 58.94	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      times-frac [=>] 37.91	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]
      unswap-sqr [=>] 11.08	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      times-frac [<=] 20.23	\[ -4 \cdot \left(\color{blue}{\frac{a \cdot b}{x-scale \cdot y-scale}} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right) \]
      *-commutative [<=] 20.23	\[ -4 \cdot \left(\frac{a \cdot b}{\color{blue}{y-scale \cdot x-scale}} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right) \]
      times-frac [<=] 10.13	\[ -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{a \cdot b}{x-scale \cdot y-scale}}\right) \]
      *-commutative [<=] 10.13	\[ -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \frac{a \cdot b}{\color{blue}{y-scale \cdot x-scale}}\right) \]
      unpow2 [<=] 10.13	\[ -4 \cdot \color{blue}{{\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2}} \]
      times-frac [=>] 10.4	\[ -4 \cdot {\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}}^{2} \]

   5. Applied egg-rr 10.39

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

4. Final simplification 9.02

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+106}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-248}:\\ \;\;\;\;\frac{\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{elif}\;b \leq 10^{-216}:\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \frac{-4 \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.64%
Cost	1616

\[\begin{array}{l} t_0 := \frac{b \cdot a}{y-scale \cdot x-scale}\\ t_1 := \frac{\left(-4 \cdot \frac{a}{x-scale}\right) \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ t_2 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ \mathbf{if}\;b \leq -3.1 \cdot 10^{+104}:\\ \;\;\;\;-4 \cdot \frac{t_2}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{elif}\;b \leq -2.4 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-235}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_2 \cdot t_2\right)\\ \end{array} \]

Alternative 2
Error	8.36%
Cost	1353

\[\begin{array}{l} t_0 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ t_1 := \frac{b \cdot a}{y-scale \cdot x-scale}\\ \mathbf{if}\;y-scale \leq -5.5 \cdot 10^{+68} \lor \neg \left(y-scale \leq 1.7 \cdot 10^{-298}\right):\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \end{array} \]

Alternative 3
Error	8.51%
Cost	1353

\[\begin{array}{l} t_0 := \frac{b \cdot a}{y-scale \cdot x-scale}\\ \mathbf{if}\;y-scale \leq -6.5 \cdot 10^{+68} \lor \neg \left(y-scale \leq 7.2 \cdot 10^{-296}\right):\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 4
Error	10.6%
Cost	1220

\[\begin{array}{l} t_0 := \frac{a}{y-scale} \cdot \frac{b}{x-scale}\\ \mathbf{if}\;a \leq 5 \cdot 10^{+151}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{y-scale} \cdot \frac{b \cdot a}{y-scale \cdot \left(x-scale \cdot \frac{x-scale}{a}\right)}\right)\\ \end{array} \]

Alternative 5
Error	8.53%
Cost	1088

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

Alternative 6
Error	47.72%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023121 
(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))))