Simplification of discriminant from scale-rotated-ellipse

Average Error: 40.6 → 6.6

Time: 1.0min

Precision: binary64

Cost: 13900

Math

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

↓

\[\begin{array}{l} t_0 := -4 \cdot {\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}\\ \mathbf{if}\;a \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{b}{y-scale} \cdot -4\right)\right)\\ \mathbf{elif}\;a \leq -7.592012976852885 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 5.536322656112811 \cdot 10^{-282}:\\ \;\;\;\;-4 \cdot {\left({\left(\frac{x-scale \cdot y-scale}{a \cdot b}\right)}^{2}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (-
  (*
   (/
    (/
     (*
      (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI)))
      (cos (* (/ angle 180.0) PI)))
     x-scale)
    y-scale)
   (/
    (/
     (*
      (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI)))
      (cos (* (/ angle 180.0) PI)))
     x-scale)
    y-scale))
  (*
   (*
    4.0
    (/
     (/
      (+
       (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
       (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
      x-scale)
     x-scale))
   (/
    (/
     (+
      (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
      (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
     y-scale)
    y-scale))))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* -4.0 (pow (* (/ a y-scale) (/ b x-scale)) 2.0))))
   (if (<= a -1e+235)
     (*
      (* (/ a x-scale) (/ a x-scale))
      (* (/ b y-scale) (* (/ b y-scale) -4.0)))
     (if (<= a -7.592012976852885e-130)
       t_0
       (if (<= a 5.536322656112811e-282)
         (* -4.0 (pow (pow (/ (* x-scale y-scale) (* a b)) 2.0) -1.0))
         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 = -4.0 * pow(((a / y_45_scale) * (b / x_45_scale)), 2.0);
	double tmp;
	if (a <= -1e+235) {
		tmp = ((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * ((b / y_45_scale) * -4.0));
	} else if (a <= -7.592012976852885e-130) {
		tmp = t_0;
	} else if (a <= 5.536322656112811e-282) {
		tmp = -4.0 * pow(pow(((x_45_scale * y_45_scale) / (a * b)), 2.0), -1.0);
	} else {
		tmp = 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 = -4.0 * Math.pow(((a / y_45_scale) * (b / x_45_scale)), 2.0);
	double tmp;
	if (a <= -1e+235) {
		tmp = ((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * ((b / y_45_scale) * -4.0));
	} else if (a <= -7.592012976852885e-130) {
		tmp = t_0;
	} else if (a <= 5.536322656112811e-282) {
		tmp = -4.0 * Math.pow(Math.pow(((x_45_scale * y_45_scale) / (a * b)), 2.0), -1.0);
	} else {
		tmp = 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 = -4.0 * math.pow(((a / y_45_scale) * (b / x_45_scale)), 2.0)
	tmp = 0
	if a <= -1e+235:
		tmp = ((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * ((b / y_45_scale) * -4.0))
	elif a <= -7.592012976852885e-130:
		tmp = t_0
	elif a <= 5.536322656112811e-282:
		tmp = -4.0 * math.pow(math.pow(((x_45_scale * y_45_scale) / (a * b)), 2.0), -1.0)
	else:
		tmp = 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(-4.0 * (Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale)) ^ 2.0))
	tmp = 0.0
	if (a <= -1e+235)
		tmp = Float64(Float64(Float64(a / x_45_scale) * Float64(a / x_45_scale)) * Float64(Float64(b / y_45_scale) * Float64(Float64(b / y_45_scale) * -4.0)));
	elseif (a <= -7.592012976852885e-130)
		tmp = t_0;
	elseif (a <= 5.536322656112811e-282)
		tmp = Float64(-4.0 * ((Float64(Float64(x_45_scale * y_45_scale) / Float64(a * b)) ^ 2.0) ^ -1.0));
	else
		tmp = 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 = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) ^ 2.0);
	tmp = 0.0;
	if (a <= -1e+235)
		tmp = ((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * ((b / y_45_scale) * -4.0));
	elseif (a <= -7.592012976852885e-130)
		tmp = t_0;
	elseif (a <= 5.536322656112811e-282)
		tmp = -4.0 * ((((x_45_scale * y_45_scale) / (a * b)) ^ 2.0) ^ -1.0);
	else
		tmp = 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[(-4.0 * N[Power[N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1e+235], N[(N[(N[(a / x$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / y$45$scale), $MachinePrecision] * N[(N[(b / y$45$scale), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -7.592012976852885e-130], t$95$0, If[LessEqual[a, 5.536322656112811e-282], N[(-4.0 * N[Power[N[Power[N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.0000000000000001e235

   1. Initial program 64.0

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

   2. Taylor expanded in angle around 0 64.0

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

   3. Simplified 30.9

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

   4. Applied egg-rr 33.0

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

   5. Taylor expanded in a around 0 64.0

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

   6. Simplified 32.4

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

   if -1.0000000000000001e235 < a < -7.59201297685288518e-130 or 5.5363226561128112e-282 < a

   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. Taylor expanded in angle around 0 38.5

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

   3. Simplified 20.6

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

   4. Applied egg-rr 5.7

      \[\leadsto \color{blue}{0 + -4 \cdot {\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}} \]

   if -7.59201297685288518e-130 < a < 5.5363226561128112e-282

   1. Initial program 32.5

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

   2. Simplified 33.6

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

   3. Taylor expanded in angle around 0 36.1

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

   4. Applied egg-rr 5.5

      \[\leadsto -4 \cdot \color{blue}{{\left({\left(\frac{y-scale \cdot x-scale}{a \cdot b}\right)}^{2}\right)}^{-1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{b}{y-scale} \cdot -4\right)\right)\\ \mathbf{elif}\;a \leq -7.592012976852885 \cdot 10^{-130}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}\\ \mathbf{elif}\;a \leq 5.536322656112811 \cdot 10^{-282}:\\ \;\;\;\;-4 \cdot {\left({\left(\frac{x-scale \cdot y-scale}{a \cdot b}\right)}^{2}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.6
Cost	7436

\[\begin{array}{l} t_0 := \frac{a \cdot b}{x-scale \cdot y-scale}\\ t_1 := -4 \cdot {\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}\\ \mathbf{if}\;a \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \left(\frac{b}{y-scale} \cdot -4\right)\right)\\ \mathbf{elif}\;a \leq -7.592012976852885 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.536322656112811 \cdot 10^{-282}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	10.7
Cost	1616

\[\begin{array}{l} t_0 := -4 \cdot \left(\frac{a}{x-scale} \cdot \frac{\frac{\frac{a}{x-scale} \cdot b}{\frac{y-scale}{b}}}{y-scale}\right)\\ \mathbf{if}\;a \leq -1 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -3.282432594993207 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{b \cdot \left(-4 \cdot \frac{a \cdot a}{x-scale \cdot y-scale}\right)}{y-scale}}{\frac{x-scale}{b}}\\ \mathbf{elif}\;a \leq 2.1080106217200678 \cdot 10^{-240}:\\ \;\;\;\;-4 \cdot \frac{a \cdot b}{\left(x-scale \cdot y-scale\right) \cdot \frac{x-scale}{\frac{a \cdot b}{y-scale}}}\\ \mathbf{elif}\;a \leq 5.568508627575101 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot \frac{y-scale}{b}}\right)\\ \end{array} \]

Alternative 3
Error	11.1
Cost	1616

\[\begin{array}{l} t_0 := \frac{\frac{b}{y-scale} \cdot -4}{\frac{x-scale}{b} \cdot \left(\frac{y-scale}{a} \cdot \frac{x-scale}{a}\right)}\\ t_1 := \frac{x-scale}{\frac{a \cdot b}{y-scale}}\\ \mathbf{if}\;a \leq -1 \cdot 10^{+96}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{\frac{\frac{a}{x-scale} \cdot b}{\frac{y-scale}{b}}}{y-scale}\right)\\ \mathbf{elif}\;a \leq -3.282432594993207 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 9.0673547687726 \cdot 10^{-191}:\\ \;\;\;\;-4 \cdot \frac{a \cdot b}{\left(x-scale \cdot y-scale\right) \cdot t_1}\\ \mathbf{elif}\;a \leq 8.382983644015173 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{t_1}\\ \end{array} \]

Alternative 4
Error	8.0
Cost	1352

\[\begin{array}{l} t_0 := \frac{a \cdot b}{y-scale}\\ t_1 := -4 \cdot \frac{t_0}{x-scale \cdot \frac{x-scale}{t_0}}\\ \mathbf{if}\;x-scale \leq -1 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq 10^{-250}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{\frac{\frac{a}{x-scale} \cdot b}{\frac{y-scale}{b}}}{y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	6.7
Cost	1352

\[\begin{array}{l} t_0 := \frac{a \cdot b}{x-scale \cdot y-scale}\\ t_1 := -4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{if}\;y-scale \leq 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 10^{-165}:\\ \;\;\;\;\frac{\frac{b}{y-scale} \cdot -4}{\frac{x-scale}{b} \cdot \left(\frac{y-scale}{a} \cdot \frac{x-scale}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	9.1
Cost	1088

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

Alternative 7
Error	30.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022217 
(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))))