Simplification of discriminant from scale-rotated-ellipse

Average Error: 40.4 → 10.7

Time: 1.4min

Precision: binary64

Cost: 1484

Math

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

↓

\[\begin{array}{l} t_0 := \frac{b \cdot a}{y-scale \cdot x-scale}\\ t_1 := -4 \cdot \left(\frac{b \cdot \frac{a}{y-scale}}{x-scale} \cdot t_0\right)\\ t_2 := \frac{y-scale}{\frac{a}{x-scale}}\\ \mathbf{if}\;b \leq 1.2667315210438861 \cdot 10^{-229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.621613892688902 \cdot 10^{-94}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{t_2} \cdot t_0\right)\\ \mathbf{elif}\;b \leq 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot b}{y-scale} \cdot \frac{b}{\frac{x-scale}{a}}}{t_2}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b a) (* y-scale x-scale)))
        (t_1 (* -4.0 (* (/ (* b (/ a y-scale)) x-scale) t_0)))
        (t_2 (/ y-scale (/ a x-scale))))
   (if (<= b 1.2667315210438861e-229)
     t_1
     (if (<= b 3.621613892688902e-94)
       (* -4.0 (* (/ b t_2) t_0))
       (if (<= b 1e-25)
         t_1
         (/ (* (/ (* -4.0 b) y-scale) (/ b (/ x-scale a))) t_2))))))

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 = (b * a) / (y_45_scale * x_45_scale);
	double t_1 = -4.0 * (((b * (a / y_45_scale)) / x_45_scale) * t_0);
	double t_2 = y_45_scale / (a / x_45_scale);
	double tmp;
	if (b <= 1.2667315210438861e-229) {
		tmp = t_1;
	} else if (b <= 3.621613892688902e-94) {
		tmp = -4.0 * ((b / t_2) * t_0);
	} else if (b <= 1e-25) {
		tmp = t_1;
	} else {
		tmp = (((-4.0 * b) / y_45_scale) * (b / (x_45_scale / a))) / t_2;
	}
	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 = (b * a) / (y_45_scale * x_45_scale);
	double t_1 = -4.0 * (((b * (a / y_45_scale)) / x_45_scale) * t_0);
	double t_2 = y_45_scale / (a / x_45_scale);
	double tmp;
	if (b <= 1.2667315210438861e-229) {
		tmp = t_1;
	} else if (b <= 3.621613892688902e-94) {
		tmp = -4.0 * ((b / t_2) * t_0);
	} else if (b <= 1e-25) {
		tmp = t_1;
	} else {
		tmp = (((-4.0 * b) / y_45_scale) * (b / (x_45_scale / a))) / t_2;
	}
	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 = (b * a) / (y_45_scale * x_45_scale)
	t_1 = -4.0 * (((b * (a / y_45_scale)) / x_45_scale) * t_0)
	t_2 = y_45_scale / (a / x_45_scale)
	tmp = 0
	if b <= 1.2667315210438861e-229:
		tmp = t_1
	elif b <= 3.621613892688902e-94:
		tmp = -4.0 * ((b / t_2) * t_0)
	elif b <= 1e-25:
		tmp = t_1
	else:
		tmp = (((-4.0 * b) / y_45_scale) * (b / (x_45_scale / a))) / t_2
	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(b * a) / Float64(y_45_scale * x_45_scale))
	t_1 = Float64(-4.0 * Float64(Float64(Float64(b * Float64(a / y_45_scale)) / x_45_scale) * t_0))
	t_2 = Float64(y_45_scale / Float64(a / x_45_scale))
	tmp = 0.0
	if (b <= 1.2667315210438861e-229)
		tmp = t_1;
	elseif (b <= 3.621613892688902e-94)
		tmp = Float64(-4.0 * Float64(Float64(b / t_2) * t_0));
	elseif (b <= 1e-25)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(-4.0 * b) / y_45_scale) * Float64(b / Float64(x_45_scale / a))) / t_2);
	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 = (b * a) / (y_45_scale * x_45_scale);
	t_1 = -4.0 * (((b * (a / y_45_scale)) / x_45_scale) * t_0);
	t_2 = y_45_scale / (a / x_45_scale);
	tmp = 0.0;
	if (b <= 1.2667315210438861e-229)
		tmp = t_1;
	elseif (b <= 3.621613892688902e-94)
		tmp = -4.0 * ((b / t_2) * t_0);
	elseif (b <= 1e-25)
		tmp = t_1;
	else
		tmp = (((-4.0 * b) / y_45_scale) * (b / (x_45_scale / a))) / t_2;
	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[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(N[(N[(b * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y$45$scale / N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.2667315210438861e-229], t$95$1, If[LessEqual[b, 3.621613892688902e-94], N[(-4.0 * N[(N[(b / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-25], t$95$1, N[(N[(N[(N[(-4.0 * b), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(b / N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.26673152104388614e-229 or 3.621613892688902e-94 < b < 1.00000000000000004e-25

   1. Initial program 38.5

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

   2. Taylor expanded in angle around 0 38.0

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

   3. Simplified 29.7

      \[\leadsto \color{blue}{-4 \cdot \frac{b \cdot b}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{a \cdot a}}} \]
      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)))): 42 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))))): 1 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))))): 7 points increase in error, 3 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. Taylor expanded in b around 0 38.0

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

   5. Simplified 24.8

      \[\leadsto -4 \cdot \color{blue}{\left(a \cdot \left(\frac{a}{y-scale} \cdot \frac{b \cdot b}{x-scale \cdot \left(y-scale \cdot x-scale\right)}\right)\right)} \]
      Proof
      (*.f64 a (*.f64 (/.f64 a y-scale) (/.f64 (*.f64 b b) (*.f64 x-scale (*.f64 y-scale x-scale))))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (*.f64 (/.f64 a y-scale) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 x-scale (*.f64 y-scale x-scale))))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (*.f64 (/.f64 a y-scale) (/.f64 (pow.f64 b 2) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 y-scale x-scale) x-scale))))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (*.f64 (/.f64 a y-scale) (/.f64 (pow.f64 b 2) (Rewrite<= associate-*r*_binary64 (*.f64 y-scale (*.f64 x-scale x-scale)))))): 15 points increase in error, 10 points decrease in error
      (*.f64 a (*.f64 (/.f64 a y-scale) (/.f64 (pow.f64 b 2) (*.f64 y-scale (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (Rewrite<= associate-/r/_binary64 (/.f64 a (/.f64 y-scale (/.f64 (pow.f64 b 2) (*.f64 y-scale (pow.f64 x-scale 2))))))): 8 points increase in error, 16 points decrease in error
      (*.f64 a (/.f64 a (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y-scale (*.f64 y-scale (pow.f64 x-scale 2))) (pow.f64 b 2))))): 11 points increase in error, 5 points decrease in error
      (*.f64 a (/.f64 a (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y-scale y-scale) (pow.f64 x-scale 2))) (pow.f64 b 2)))): 29 points increase in error, 3 points decrease in error
      (*.f64 a (/.f64 a (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2)) (pow.f64 x-scale 2)) (pow.f64 b 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 a (/.f64 (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)) (pow.f64 b 2))) a)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 a a) (/.f64 (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)) (pow.f64 b 2)))): 28 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (/.f64 (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)) (pow.f64 b 2))): 0 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)))): 12 points increase in error, 2 points decrease in error

   6. Applied egg-rr 20.5

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

   7. Applied egg-rr 10.0

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

   if 1.26673152104388614e-229 < b < 3.621613892688902e-94

   1. Initial program 32.2

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

   2. Taylor expanded in angle around 0 34.8

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

   3. Simplified 26.2

      \[\leadsto \color{blue}{-4 \cdot \frac{b \cdot b}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{a \cdot a}}} \]
      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)))): 42 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))))): 1 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))))): 7 points increase in error, 3 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 3.8

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

   5. Applied egg-rr 3.7

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

   6. Taylor expanded in b around -inf 8.0

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

   if 1.00000000000000004e-25 < b

   1. Initial program 51.4

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

   2. Taylor expanded in angle around 0 45.6

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

   3. Simplified 38.4

      \[\leadsto \color{blue}{-4 \cdot \frac{b \cdot b}{\frac{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}{a \cdot a}}} \]
      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)))): 42 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))))): 1 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))))): 7 points increase in error, 3 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 26.2

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

   5. Applied egg-rr 14.6

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

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2667315210438861 \cdot 10^{-229}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot \frac{a}{y-scale}}{x-scale} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{elif}\;b \leq 3.621613892688902 \cdot 10^{-94}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{\frac{y-scale}{\frac{a}{x-scale}}} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{elif}\;b \leq 10^{-25}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot \frac{a}{y-scale}}{x-scale} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot b}{y-scale} \cdot \frac{b}{\frac{x-scale}{a}}}{\frac{y-scale}{\frac{a}{x-scale}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.9
Cost	1484

\[\begin{array}{l} t_0 := -4 \cdot \left(a \cdot \frac{a \cdot \frac{b}{x-scale}}{y-scale \cdot \left(x-scale \cdot \frac{y-scale}{b}\right)}\right)\\ t_1 := -4 \cdot \frac{a \cdot \left(b \cdot \frac{a}{\frac{y-scale}{b}}\right)}{x-scale \cdot \left(y-scale \cdot x-scale\right)}\\ \mathbf{if}\;y-scale \leq -1.5593015565557055 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -1 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 5 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.2
Cost	1352

\[\begin{array}{l} t_0 := -4 \cdot \left(\frac{b}{\frac{y-scale}{\frac{a}{x-scale}}} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{if}\;b \leq -1.6906411120715255 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.2667315210438861 \cdot 10^{-229}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{a \cdot \frac{b}{x-scale}}{y-scale \cdot \left(x-scale \cdot \frac{y-scale}{b}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.4
Cost	1220

\[\begin{array}{l} t_0 := \frac{b \cdot a}{y-scale \cdot x-scale}\\ \mathbf{if}\;b \leq 1.2667315210438861 \cdot 10^{-229}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot \frac{a}{y-scale}}{x-scale} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{\frac{y-scale}{\frac{a}{x-scale}}} \cdot t_0\right)\\ \end{array} \]

Alternative 4
Error	8.5
Cost	1220

\[\begin{array}{l} \mathbf{if}\;b \leq 1.2667315210438861 \cdot 10^{-229}:\\ \;\;\;\;-4 \cdot \left(\frac{b \cdot \frac{a}{y-scale}}{x-scale} \cdot \frac{b \cdot a}{y-scale \cdot x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{\frac{y-scale}{\frac{a}{x-scale}}} \cdot \frac{b}{y-scale \cdot \frac{x-scale}{a}}\right)\\ \end{array} \]

Alternative 5
Error	19.7
Cost	1088

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

Alternative 6
Error	6.0
Cost	1088

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

Alternative 7
Error	29.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022311 
(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))))