Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.4 → 6.0

Time: 1.3min

Precision: binary64

Cost: 7436

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\\ t_1 := -4 \cdot {\left(\frac{b \cdot \frac{a}{x-scale}}{y-scale}\right)}^{2}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-195}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-4 \cdot t_0\right)\\ \end{array} \]

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 (/ (/ a y-scale) (/ x-scale b)))
        (t_1 (* -4.0 (pow (/ (* b (/ a x-scale)) y-scale) 2.0))))
   (if (<= b -2e-135)
     t_1
     (if (<= b 5e-195)
       (*
        -4.0
        (/ (* (/ a y-scale) (/ b x-scale)) (* (/ y-scale a) (/ x-scale b))))
       (if (<= b 2.9e-24) t_1 (* t_0 (* -4.0 t_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 = (a / y_45_scale) / (x_45_scale / b);
	double t_1 = -4.0 * pow(((b * (a / x_45_scale)) / y_45_scale), 2.0);
	double tmp;
	if (b <= -2e-135) {
		tmp = t_1;
	} else if (b <= 5e-195) {
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((y_45_scale / a) * (x_45_scale / b)));
	} else if (b <= 2.9e-24) {
		tmp = t_1;
	} else {
		tmp = t_0 * (-4.0 * t_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 = (a / y_45_scale) / (x_45_scale / b);
	double t_1 = -4.0 * Math.pow(((b * (a / x_45_scale)) / y_45_scale), 2.0);
	double tmp;
	if (b <= -2e-135) {
		tmp = t_1;
	} else if (b <= 5e-195) {
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((y_45_scale / a) * (x_45_scale / b)));
	} else if (b <= 2.9e-24) {
		tmp = t_1;
	} else {
		tmp = t_0 * (-4.0 * t_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 = (a / y_45_scale) / (x_45_scale / b)
	t_1 = -4.0 * math.pow(((b * (a / x_45_scale)) / y_45_scale), 2.0)
	tmp = 0
	if b <= -2e-135:
		tmp = t_1
	elif b <= 5e-195:
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((y_45_scale / a) * (x_45_scale / b)))
	elif b <= 2.9e-24:
		tmp = t_1
	else:
		tmp = t_0 * (-4.0 * t_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(Float64(a / y_45_scale) / Float64(x_45_scale / b))
	t_1 = Float64(-4.0 * (Float64(Float64(b * Float64(a / x_45_scale)) / y_45_scale) ^ 2.0))
	tmp = 0.0
	if (b <= -2e-135)
		tmp = t_1;
	elseif (b <= 5e-195)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale)) / Float64(Float64(y_45_scale / a) * Float64(x_45_scale / b))));
	elseif (b <= 2.9e-24)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(-4.0 * t_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 = (a / y_45_scale) / (x_45_scale / b);
	t_1 = -4.0 * (((b * (a / x_45_scale)) / y_45_scale) ^ 2.0);
	tmp = 0.0;
	if (b <= -2e-135)
		tmp = t_1;
	elseif (b <= 5e-195)
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((y_45_scale / a) * (x_45_scale / b)));
	elseif (b <= 2.9e-24)
		tmp = t_1;
	else
		tmp = t_0 * (-4.0 * t_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[(N[(a / y$45$scale), $MachinePrecision] / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[Power[N[(N[(b * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e-135], t$95$1, If[LessEqual[b, 5e-195], N[(-4.0 * N[(N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale / a), $MachinePrecision] * N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e-24], t$95$1, N[(t$95$0 * N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e-135 or 5.00000000000000009e-195 < b < 2.8999999999999999e-24

   1. Initial program 41.9

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

   2. Simplified 46.0

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

      [Start] 41.9	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
      sub-neg [=>] 41.9	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]
      +-commutative [=>] 41.9	\[ \color{blue}{\left(-\left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}} \]

   3. Taylor expanded in angle around 0 38.7

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

   4. Simplified 36.0

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

      [Start] 38.7	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 38.9	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      *-commutative [=>] 38.9	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 38.9	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      associate-/l* [=>] 36.0	\[ -4 \cdot \left(\color{blue}{\frac{b}{\frac{{y-scale}^{2}}{b}}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 36.0	\[ -4 \cdot \left(\frac{b}{\frac{\color{blue}{y-scale \cdot y-scale}}{b}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 36.0	\[ -4 \cdot \left(\frac{b}{\frac{y-scale \cdot y-scale}{b}} \cdot \frac{\color{blue}{a \cdot a}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 36.0	\[ -4 \cdot \left(\frac{b}{\frac{y-scale \cdot y-scale}{b}} \cdot \frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}}\right) \]

   5. Taylor expanded in b around 0 38.7

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

   6. Simplified 5.7

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

      [Start] 38.7	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 38.9	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 38.9	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 38.9	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      times-frac [=>] 30.4	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 30.4	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 30.4	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      associate-/r* [=>] 26.6	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\frac{\frac{b \cdot b}{y-scale}}{y-scale}}\right) \]
      associate-*r/ [<=] 21.6	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{\color{blue}{b \cdot \frac{b}{y-scale}}}{y-scale}\right) \]
      associate-*l/ [<=] 20.1	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]
      swap-sqr [<=] 5.8	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
      *-commutative [<=] 5.8	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right) \]
      *-commutative [<=] 5.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}\right) \]
      unpow2 [<=] 5.8	\[ -4 \cdot \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \]
      associate-*l/ [=>] 5.7	\[ -4 \cdot {\color{blue}{\left(\frac{b \cdot \frac{a}{x-scale}}{y-scale}\right)}}^{2} \]

   if -2.0000000000000001e-135 < b < 5.00000000000000009e-195

   1. Initial program 33.6

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

   2. Taylor expanded in angle around 0 37.4

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

   3. Taylor expanded in x-scale around 0 37.4

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

   4. Simplified 5.2

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

      [Start] 37.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [<=] 37.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      times-frac [=>] 37.6	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 37.6	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 37.6	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 29.8	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 29.8	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 29.8	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 15.4	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      unswap-sqr [=>] 5.2	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      times-frac [<=] 10.3	\[ -4 \cdot \left(\color{blue}{\frac{a \cdot b}{y-scale \cdot x-scale}} \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right) \]
      times-frac [<=] 5.5	\[ -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{a \cdot b}{y-scale \cdot x-scale}}\right) \]
      unpow2 [<=] 5.5	\[ -4 \cdot \color{blue}{{\left(\frac{a \cdot b}{y-scale \cdot x-scale}\right)}^{2}} \]
      times-frac [=>] 5.2	\[ -4 \cdot {\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}}^{2} \]

   5. Applied egg-rr 5.4

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

   if 2.8999999999999999e-24 < b

   1. Initial program 51.8

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

   2. Taylor expanded in angle around 0 45.5

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

   3. Applied egg-rr 7.2

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

   4. Applied egg-rr 13.1

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

   5. Applied egg-rr 7.6

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

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-135}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot \frac{a}{x-scale}}{y-scale}\right)}^{2}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-195}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;-4 \cdot {\left(\frac{b \cdot \frac{a}{x-scale}}{y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}} \cdot \left(-4 \cdot \frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.2
Cost	1880

\[\begin{array}{l} t_0 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ t_1 := -4 \cdot \left(t_0 \cdot t_0\right)\\ t_2 := -4 \cdot \frac{a \cdot \frac{b}{x-scale}}{y-scale \cdot \left(\frac{y-scale}{a} \cdot \frac{x-scale}{b}\right)}\\ t_3 := -4 \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{a}{x-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{if}\;y-scale \leq -9.5 \cdot 10^{+215}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y-scale \leq -6.6 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 1.1 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 9 \cdot 10^{-164}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\right)\\ \mathbf{elif}\;y-scale \leq 3 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 1.15 \cdot 10^{+203}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	7.3
Cost	1880

\[\begin{array}{l} t_0 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ t_1 := -4 \cdot \frac{a \cdot \frac{b}{x-scale}}{y-scale \cdot \left(\frac{y-scale}{a} \cdot \frac{x-scale}{b}\right)}\\ t_2 := -4 \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{a}{x-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{if}\;y-scale \leq -2 \cdot 10^{+217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq -2.5 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 8 \cdot 10^{-238}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;y-scale \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 1.95 \cdot 10^{+239}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(b \cdot \frac{a}{y-scale}\right)}{x-scale}\\ \end{array} \]

Alternative 3
Error	7.5
Cost	1485

\[\begin{array}{l} t_0 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ \mathbf{if}\;y-scale \leq -4.2 \cdot 10^{+178} \lor \neg \left(y-scale \leq 1.1 \cdot 10^{-257}\right) \land y-scale \leq 1.25 \cdot 10^{+203}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{a}{x-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 4
Error	8.4
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -4 \cdot 10^{-8} \lor \neg \left(x-scale \leq -3.2 \cdot 10^{-175}\right):\\ \;\;\;\;-4 \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{a}{x-scale} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot a}{x-scale \cdot y-scale} \cdot \left(a \cdot \frac{b}{x-scale \cdot y-scale}\right)\right)\\ \end{array} \]

Alternative 5
Error	5.8
Cost	1353

\[\begin{array}{l} t_0 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ \mathbf{if}\;angle \leq 1.45 \cdot 10^{-163} \lor \neg \left(angle \leq 7.2 \cdot 10^{-97}\right):\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 6
Error	5.7
Cost	1352

\[\begin{array}{l} t_0 := \frac{b \cdot a}{x-scale \cdot y-scale}\\ t_1 := \frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\\ \mathbf{if}\;angle \leq 5.6 \cdot 10^{-157}:\\ \;\;\;\;t_1 \cdot \left(-4 \cdot t_1\right)\\ \mathbf{elif}\;angle \leq 6.2 \cdot 10^{-97}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{y-scale}{a} \cdot \frac{x-scale}{b}}\\ \end{array} \]

Alternative 7
Error	8.5
Cost	1088

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

Alternative 8
Error	30.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023187 
(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))))