Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.6 → 5.6

Time: 1.3min

Precision: binary64

Cost: 13636

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a \cdot \frac{b}{x-scale}}{y-scale}\\ \mathbf{if}\;b \leq 6.632805862188225 \cdot 10^{-195}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left({\left(\frac{\frac{y-scale}{\frac{a}{x-scale}}}{b}\right)}^{2}\right)}^{-1}\\ \end{array} \]

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 (/ b x-scale)) y-scale)))
   (if (<= b 6.632805862188225e-195)
     (* -4.0 (* t_0 t_0))
     (* -4.0 (pow (pow (/ (/ y-scale (/ a x-scale)) b) 2.0) -1.0)))))

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 * (b / x_45_scale)) / y_45_scale;
	double tmp;
	if (b <= 6.632805862188225e-195) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * pow(pow(((y_45_scale / (a / x_45_scale)) / b), 2.0), -1.0);
	}
	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 * (b / x_45_scale)) / y_45_scale;
	double tmp;
	if (b <= 6.632805862188225e-195) {
		tmp = -4.0 * (t_0 * t_0);
	} else {
		tmp = -4.0 * Math.pow(Math.pow(((y_45_scale / (a / x_45_scale)) / b), 2.0), -1.0);
	}
	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 * (b / x_45_scale)) / y_45_scale
	tmp = 0
	if b <= 6.632805862188225e-195:
		tmp = -4.0 * (t_0 * t_0)
	else:
		tmp = -4.0 * math.pow(math.pow(((y_45_scale / (a / x_45_scale)) / b), 2.0), -1.0)
	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 * Float64(b / x_45_scale)) / y_45_scale)
	tmp = 0.0
	if (b <= 6.632805862188225e-195)
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	else
		tmp = Float64(-4.0 * ((Float64(Float64(y_45_scale / Float64(a / x_45_scale)) / b) ^ 2.0) ^ -1.0));
	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 * (b / x_45_scale)) / y_45_scale;
	tmp = 0.0;
	if (b <= 6.632805862188225e-195)
		tmp = -4.0 * (t_0 * t_0);
	else
		tmp = -4.0 * ((((y_45_scale / (a / x_45_scale)) / b) ^ 2.0) ^ -1.0);
	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 * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[b, 6.632805862188225e-195], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[Power[N[(N[(y$45$scale / N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.63280586218822505e-195

   1. Initial program 40.1

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

   2. Taylor expanded in angle around 0 39.2

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

   3. Simplified 31.1

      \[\leadsto \color{blue}{-4 \cdot \frac{b \cdot b}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{a \cdot a}}} \]
      Proof
      (*.f64 -4 (/.f64 (*.f64 b b) (/.f64 (*.f64 (*.f64 y-scale x-scale) (*.f64 y-scale x-scale)) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (/.f64 (*.f64 (*.f64 y-scale x-scale) (*.f64 y-scale x-scale)) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (/.f64 (pow.f64 b 2) (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 y-scale y-scale) (*.f64 x-scale x-scale))) (*.f64 a a)))): 36 points increase in error, 1 points decrease in error
      (*.f64 -4 (/.f64 (pow.f64 b 2) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2)) (*.f64 x-scale x-scale)) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (/.f64 (pow.f64 b 2) (/.f64 (*.f64 (pow.f64 y-scale 2) (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2))) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (/.f64 (pow.f64 b 2) (/.f64 (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)) (Rewrite<= unpow2_binary64 (pow.f64 a 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 a 2)) (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2))))): 3 points increase in error, 4 points decrease in error
      (*.f64 -4 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 a 2) (pow.f64 b 2))) (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 6.4

      \[\leadsto -4 \cdot \color{blue}{{\left({\left(\frac{\frac{y-scale}{\frac{a}{x-scale}}}{b}\right)}^{2}\right)}^{-1}} \]

   5. Taylor expanded in y-scale around 0 39.2

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

   6. Simplified 5.8

      \[\leadsto -4 \cdot \color{blue}{\left(\frac{a \cdot \frac{b}{x-scale}}{y-scale} \cdot \frac{a \cdot \frac{b}{x-scale}}{y-scale}\right)} \]
      Proof
      (*.f64 (/.f64 (*.f64 a (/.f64 b x-scale)) y-scale) (/.f64 (*.f64 a (/.f64 b x-scale)) y-scale)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 a y-scale) (/.f64 b x-scale))) (/.f64 (*.f64 a (/.f64 b x-scale)) y-scale)): 15 points increase in error, 21 points decrease in error
      (*.f64 (*.f64 (/.f64 a y-scale) (/.f64 b x-scale)) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 a y-scale) (/.f64 b x-scale)))): 14 points increase in error, 21 points decrease in error
      (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (/.f64 a y-scale) (/.f64 a y-scale)) (*.f64 (/.f64 b x-scale) (/.f64 b x-scale)))): 80 points increase in error, 25 points decrease in error
      (*.f64 (*.f64 (/.f64 a y-scale) (/.f64 a y-scale)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 b b) (*.f64 x-scale x-scale)))): 50 points increase in error, 15 points decrease in error
      (*.f64 (*.f64 (/.f64 a y-scale) (/.f64 a y-scale)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 x-scale x-scale))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 a y-scale) (/.f64 a y-scale)) (/.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a a) (*.f64 y-scale y-scale))) (/.f64 (pow.f64 b 2) (pow.f64 x-scale 2))): 39 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 y-scale y-scale)) (/.f64 (pow.f64 b 2) (pow.f64 x-scale 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2))) (/.f64 (pow.f64 b 2) (pow.f64 x-scale 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)))): 6 points increase in error, 17 points decrease in error

   if 6.63280586218822505e-195 < b

   1. Initial program 44.1

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

   2. Taylor expanded in angle around 0 41.2

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

   3. Simplified 32.4

      \[\leadsto \color{blue}{-4 \cdot \frac{b \cdot b}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{a \cdot a}}} \]
      Proof
      (*.f64 -4 (/.f64 (*.f64 b b) (/.f64 (*.f64 (*.f64 y-scale x-scale) (*.f64 y-scale x-scale)) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (/.f64 (*.f64 (*.f64 y-scale x-scale) (*.f64 y-scale x-scale)) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (/.f64 (pow.f64 b 2) (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 y-scale y-scale) (*.f64 x-scale x-scale))) (*.f64 a a)))): 36 points increase in error, 1 points decrease in error
      (*.f64 -4 (/.f64 (pow.f64 b 2) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2)) (*.f64 x-scale x-scale)) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (/.f64 (pow.f64 b 2) (/.f64 (*.f64 (pow.f64 y-scale 2) (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2))) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (/.f64 (pow.f64 b 2) (/.f64 (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)) (Rewrite<= unpow2_binary64 (pow.f64 a 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 a 2)) (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2))))): 3 points increase in error, 4 points decrease in error
      (*.f64 -4 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 a 2) (pow.f64 b 2))) (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 5.3

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

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.632805862188225 \cdot 10^{-195}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot \frac{b}{x-scale}}{y-scale} \cdot \frac{a \cdot \frac{b}{x-scale}}{y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left({\left(\frac{\frac{y-scale}{\frac{a}{x-scale}}}{b}\right)}^{2}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.7
Cost	1352

\[\begin{array}{l} t_0 := -4 \cdot \left(\frac{a \cdot \frac{b}{x-scale}}{y-scale} \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \mathbf{if}\;b \leq 8.281799595399174 \cdot 10^{-238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 10^{+34}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{\frac{a}{y-scale}}{x-scale}}{\frac{x-scale \cdot y-scale}{b \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.8
Cost	1352

\[\begin{array}{l} t_0 := \frac{a \cdot \frac{b}{x-scale}}{y-scale}\\ \mathbf{if}\;b \leq 1.2859562513628667 \cdot 10^{-210}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{elif}\;b \leq 10^{+34}:\\ \;\;\;\;-4 \cdot \frac{b \cdot \frac{\frac{a}{y-scale}}{x-scale}}{\frac{x-scale \cdot y-scale}{b \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot \left(\frac{b}{x-scale} \cdot \frac{a}{y-scale}\right)\right)\\ \end{array} \]

Alternative 3
Error	5.5
Cost	1220

\[\begin{array}{l} t_0 := \frac{b}{\frac{y-scale}{\frac{a}{x-scale}}}\\ t_1 := \frac{a \cdot \frac{b}{x-scale}}{y-scale}\\ \mathbf{if}\;b \leq 6.632805862188225 \cdot 10^{-195}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 4
Error	12.8
Cost	1088

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

Alternative 5
Error	7.3
Cost	1088

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

Alternative 6
Error	31.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022308 
(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))))