Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.0 → 5.6

Time: 1.2min

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}\\ t_1 := \frac{a \cdot b}{x-scale \cdot y-scale}\\ t_2 := -4 \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{if}\;a \leq 4 \cdot 10^{-262}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-200}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+22}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{t_0}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+88}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale} \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale \cdot \frac{y-scale}{b}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}}{t_0}\\ \end{array} \]

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 (/ (/ x-scale b) (/ a y-scale)))
        (t_1 (/ (* a b) (* x-scale y-scale)))
        (t_2 (* -4.0 (* t_1 t_1))))
   (if (<= a 4e-262)
     t_2
     (if (<= a 2e-200)
       (* -4.0 (pow (* (/ b y-scale) (/ a x-scale)) 2.0))
       (if (<= a 9.6e+22)
         (* -4.0 (/ (* (/ a y-scale) (/ b x-scale)) t_0))
         (if (<= a 6.8e+70)
           t_2
           (if (<= a 1.25e+88)
             (*
              -4.0
              (*
               (/ a x-scale)
               (* a (/ (/ b y-scale) (* x-scale (/ y-scale b))))))
             (* -4.0 (/ (/ (/ b x-scale) (/ y-scale a)) 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 = (x_45_scale / b) / (a / y_45_scale);
	double t_1 = (a * b) / (x_45_scale * y_45_scale);
	double t_2 = -4.0 * (t_1 * t_1);
	double tmp;
	if (a <= 4e-262) {
		tmp = t_2;
	} else if (a <= 2e-200) {
		tmp = -4.0 * pow(((b / y_45_scale) * (a / x_45_scale)), 2.0);
	} else if (a <= 9.6e+22) {
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / t_0);
	} else if (a <= 6.8e+70) {
		tmp = t_2;
	} else if (a <= 1.25e+88) {
		tmp = -4.0 * ((a / x_45_scale) * (a * ((b / y_45_scale) / (x_45_scale * (y_45_scale / b)))));
	} else {
		tmp = -4.0 * (((b / x_45_scale) / (y_45_scale / a)) / 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 = (x_45_scale / b) / (a / y_45_scale);
	double t_1 = (a * b) / (x_45_scale * y_45_scale);
	double t_2 = -4.0 * (t_1 * t_1);
	double tmp;
	if (a <= 4e-262) {
		tmp = t_2;
	} else if (a <= 2e-200) {
		tmp = -4.0 * Math.pow(((b / y_45_scale) * (a / x_45_scale)), 2.0);
	} else if (a <= 9.6e+22) {
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / t_0);
	} else if (a <= 6.8e+70) {
		tmp = t_2;
	} else if (a <= 1.25e+88) {
		tmp = -4.0 * ((a / x_45_scale) * (a * ((b / y_45_scale) / (x_45_scale * (y_45_scale / b)))));
	} else {
		tmp = -4.0 * (((b / x_45_scale) / (y_45_scale / a)) / 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 = (x_45_scale / b) / (a / y_45_scale)
	t_1 = (a * b) / (x_45_scale * y_45_scale)
	t_2 = -4.0 * (t_1 * t_1)
	tmp = 0
	if a <= 4e-262:
		tmp = t_2
	elif a <= 2e-200:
		tmp = -4.0 * math.pow(((b / y_45_scale) * (a / x_45_scale)), 2.0)
	elif a <= 9.6e+22:
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / t_0)
	elif a <= 6.8e+70:
		tmp = t_2
	elif a <= 1.25e+88:
		tmp = -4.0 * ((a / x_45_scale) * (a * ((b / y_45_scale) / (x_45_scale * (y_45_scale / b)))))
	else:
		tmp = -4.0 * (((b / x_45_scale) / (y_45_scale / a)) / 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(x_45_scale / b) / Float64(a / y_45_scale))
	t_1 = Float64(Float64(a * b) / Float64(x_45_scale * y_45_scale))
	t_2 = Float64(-4.0 * Float64(t_1 * t_1))
	tmp = 0.0
	if (a <= 4e-262)
		tmp = t_2;
	elseif (a <= 2e-200)
		tmp = Float64(-4.0 * (Float64(Float64(b / y_45_scale) * Float64(a / x_45_scale)) ^ 2.0));
	elseif (a <= 9.6e+22)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale)) / t_0));
	elseif (a <= 6.8e+70)
		tmp = t_2;
	elseif (a <= 1.25e+88)
		tmp = Float64(-4.0 * Float64(Float64(a / x_45_scale) * Float64(a * Float64(Float64(b / y_45_scale) / Float64(x_45_scale * Float64(y_45_scale / b))))));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(b / x_45_scale) / Float64(y_45_scale / a)) / 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 = (x_45_scale / b) / (a / y_45_scale);
	t_1 = (a * b) / (x_45_scale * y_45_scale);
	t_2 = -4.0 * (t_1 * t_1);
	tmp = 0.0;
	if (a <= 4e-262)
		tmp = t_2;
	elseif (a <= 2e-200)
		tmp = -4.0 * (((b / y_45_scale) * (a / x_45_scale)) ^ 2.0);
	elseif (a <= 9.6e+22)
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / t_0);
	elseif (a <= 6.8e+70)
		tmp = t_2;
	elseif (a <= 1.25e+88)
		tmp = -4.0 * ((a / x_45_scale) * (a * ((b / y_45_scale) / (x_45_scale * (y_45_scale / b)))));
	else
		tmp = -4.0 * (((b / x_45_scale) / (y_45_scale / a)) / 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[(x$45$scale / b), $MachinePrecision] / N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 4e-262], t$95$2, If[LessEqual[a, 2e-200], N[(-4.0 * N[Power[N[(N[(b / y$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.6e+22], N[(-4.0 * N[(N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.8e+70], t$95$2, If[LessEqual[a, 1.25e+88], N[(-4.0 * N[(N[(a / x$45$scale), $MachinePrecision] * N[(a * N[(N[(b / y$45$scale), $MachinePrecision] / N[(x$45$scale * N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(b / x$45$scale), $MachinePrecision] / N[(y$45$scale / a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if a < 4.00000000000000005e-262 or 9.6e22 < a < 6.8000000000000002e70

   1. Initial program 40.7

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

   2. Simplified 44.5

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

      [Start] 40.7

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

   3. Taylor expanded in angle around 0 38.9

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

   4. Simplified 26.8

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

      [Start] 38.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 38.9	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 39.0	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 39.0	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 39.0	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 31.0	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 31.0	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 26.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 26.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Taylor expanded in b around 0 38.9

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

   6. Simplified 31.0

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

      [Start] 38.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [<=] 38.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      associate-/r* [=>] 38.6	\[ -4 \cdot \color{blue}{\frac{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2}}}{{x-scale}^{2}}} \]
      *-commutative [=>] 38.6	\[ -4 \cdot \frac{\frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2}}}{{x-scale}^{2}} \]
      unpow2 [=>] 38.6	\[ -4 \cdot \frac{\frac{\color{blue}{\left(b \cdot b\right)} \cdot {a}^{2}}{{y-scale}^{2}}}{{x-scale}^{2}} \]
      unpow2 [=>] 38.6	\[ -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{y-scale}^{2}}}{{x-scale}^{2}} \]
      unpow2 [=>] 38.6	\[ -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{y-scale \cdot y-scale}}}{{x-scale}^{2}} \]
      unswap-sqr [=>] 31.0	\[ -4 \cdot \frac{\frac{\color{blue}{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}}{y-scale \cdot y-scale}}{{x-scale}^{2}} \]
      unpow2 [=>] 31.0	\[ -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot y-scale}}{\color{blue}{x-scale \cdot x-scale}} \]

   7. Applied egg-rr 5.6

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

   if 4.00000000000000005e-262 < a < 2e-200

   1. Initial program 30.7

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

   2. Simplified 38.6

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

      [Start] 30.7

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

   3. Taylor expanded in angle around 0 36.4

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

   4. Simplified 22.8

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

      [Start] 36.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 36.4	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 36.9	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 36.9	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 36.9	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 27.1	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 27.1	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 22.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 22.8	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Taylor expanded in b around 0 36.4

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

   6. Simplified 4.6

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

      [Start] 36.4	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      times-frac [=>] 36.9	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
      unpow2 [=>] 36.9	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 36.9	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      associate-*l/ [<=] 35.3	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale \cdot x-scale} \cdot a\right)} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 35.3	\[ -4 \cdot \left(\left(\frac{a}{x-scale \cdot x-scale} \cdot a\right) \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
      unpow2 [=>] 35.3	\[ -4 \cdot \left(\left(\frac{a}{x-scale \cdot x-scale} \cdot a\right) \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]
      *-commutative [=>] 35.3	\[ -4 \cdot \left(\color{blue}{\left(a \cdot \frac{a}{x-scale \cdot x-scale}\right)} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      associate-/r* [=>] 29.9	\[ -4 \cdot \left(\left(a \cdot \color{blue}{\frac{\frac{a}{x-scale}}{x-scale}}\right) \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      associate-*r/ [=>] 28.9	\[ -4 \cdot \left(\color{blue}{\frac{a \cdot \frac{a}{x-scale}}{x-scale}} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      associate-*l/ [<=] 28.0	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      unpow2 [<=] 28.0	\[ -4 \cdot \left(\color{blue}{{\left(\frac{a}{x-scale}\right)}^{2}} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
      associate-*r/ [=>] 27.1	\[ -4 \cdot \color{blue}{\frac{{\left(\frac{a}{x-scale}\right)}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot y-scale}} \]
      times-frac [=>] 19.6	\[ -4 \cdot \color{blue}{\left(\frac{{\left(\frac{a}{x-scale}\right)}^{2}}{y-scale} \cdot \frac{b \cdot b}{y-scale}\right)} \]
      associate-*l/ [<=] 14.6	\[ -4 \cdot \left(\frac{{\left(\frac{a}{x-scale}\right)}^{2}}{y-scale} \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot b\right)}\right) \]
      associate-/r/ [<=] 14.6	\[ -4 \cdot \left(\frac{{\left(\frac{a}{x-scale}\right)}^{2}}{y-scale} \cdot \color{blue}{\frac{b}{\frac{y-scale}{b}}}\right) \]
      associate-*l/ [=>] 14.0	\[ -4 \cdot \color{blue}{\frac{{\left(\frac{a}{x-scale}\right)}^{2} \cdot \frac{b}{\frac{y-scale}{b}}}{y-scale}} \]
      associate-*r/ [<=] 15.9	\[ -4 \cdot \color{blue}{\left({\left(\frac{a}{x-scale}\right)}^{2} \cdot \frac{\frac{b}{\frac{y-scale}{b}}}{y-scale}\right)} \]
      *-commutative [=>] 15.9	\[ -4 \cdot \color{blue}{\left(\frac{\frac{b}{\frac{y-scale}{b}}}{y-scale} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right)} \]
      associate-/r/ [=>] 16.0	\[ -4 \cdot \left(\frac{\color{blue}{\frac{b}{y-scale} \cdot b}}{y-scale} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right) \]
      associate-*r/ [<=] 12.7	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right) \]
      unpow2 [=>] 12.7	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right) \]

   if 2e-200 < a < 9.6e22

   1. Initial program 34.8

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

   2. Taylor expanded in angle around 0 32.9

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

   3. Simplified 16.4

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

      [Start] 32.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 32.9	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 33.2	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 33.2	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 33.2	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 33.2	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 29.0	\[ \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 29.0	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 29.0	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot -4\right) \]
      times-frac [=>] 16.4	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Taylor expanded in a around 0 32.9

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

   5. Simplified 3.4

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

      [Start] 32.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 32.9	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      times-frac [=>] 33.2	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 33.2	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 33.2	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 29.0	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 29.0	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 29.0	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 16.4	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 3.5	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      unpow2 [<=] 3.5	\[ -4 \cdot \color{blue}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}} \]
      associate-*r/ [=>] 3.5	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}}^{2} \]
      associate-/l* [=>] 3.4	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}}^{2} \]

   6. Applied egg-rr 3.5

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

   if 6.8000000000000002e70 < a < 1.24999999999999999e88

   1. Initial program 43.1

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

   2. Simplified 45.6

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

      [Start] 43.1

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

   3. Taylor expanded in angle around 0 34.6

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

   4. Simplified 26.2

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

      [Start] 34.6	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 34.6	\[ -4 \cdot \frac{\color{blue}{{b}^{2} \cdot {a}^{2}}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      times-frac [=>] 37.1	\[ -4 \cdot \color{blue}{\left(\frac{{b}^{2}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 37.1	\[ -4 \cdot \left(\frac{\color{blue}{b \cdot b}}{{y-scale}^{2}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 37.1	\[ -4 \cdot \left(\frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 27.5	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{{a}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 27.5	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{{a}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      associate-/r* [=>] 26.2	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \color{blue}{\frac{\frac{{a}^{2}}{x-scale}}{x-scale}}\right) \]
      unpow2 [=>] 26.2	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{\color{blue}{a \cdot a}}{x-scale}}{x-scale}\right) \]

   5. Applied egg-rr 14.5

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

   6. Applied egg-rr 12.7

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

   if 1.24999999999999999e88 < a

   1. Initial program 58.5

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

   2. Taylor expanded in angle around 0 54.8

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

   3. Simplified 25.5

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

      [Start] 54.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \]
      *-commutative [=>] 54.8	\[ \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} \cdot -4} \]
      times-frac [=>] 54.7	\[ \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \cdot -4 \]
      associate-*l* [=>] 54.7	\[ \color{blue}{\frac{{a}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right)} \]
      unpow2 [=>] 54.7	\[ \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 54.7	\[ \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      times-frac [=>] 38.3	\[ \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 38.3	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{\color{blue}{b \cdot b}}{{x-scale}^{2}} \cdot -4\right) \]
      unpow2 [=>] 38.3	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}} \cdot -4\right) \]
      times-frac [=>] 25.5	\[ \left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)} \cdot -4\right) \]

   4. Taylor expanded in a around 0 54.8

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

   5. Simplified 9.5

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

      [Start] 54.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
      *-commutative [=>] 54.8	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]
      times-frac [=>] 54.7	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)} \]
      unpow2 [=>] 54.7	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{y-scale}^{2}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 54.7	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      times-frac [=>] 38.3	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right)} \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 38.3	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{\color{blue}{b \cdot b}}{{x-scale}^{2}}\right) \]
      unpow2 [=>] 38.3	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \frac{b \cdot b}{\color{blue}{x-scale \cdot x-scale}}\right) \]
      times-frac [=>] 25.4	\[ -4 \cdot \left(\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \color{blue}{\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)}\right) \]
      swap-sqr [<=] 9.4	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right) \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)} \]
      unpow2 [<=] 9.4	\[ -4 \cdot \color{blue}{{\left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}^{2}} \]
      associate-*r/ [=>] 10.6	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale} \cdot b}{x-scale}\right)}}^{2} \]
      associate-/l* [=>] 9.5	\[ -4 \cdot {\color{blue}{\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}}^{2} \]

   6. Applied egg-rr 9.5

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

   7. Applied egg-rr 9.4

      \[\leadsto -4 \cdot \frac{\color{blue}{\frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{-262}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-200}:\\ \;\;\;\;-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+22}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+70}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \frac{a \cdot b}{x-scale \cdot y-scale}\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+88}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale} \cdot \left(a \cdot \frac{\frac{b}{y-scale}}{x-scale \cdot \frac{y-scale}{b}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{\frac{b}{x-scale}}{\frac{y-scale}{a}}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.9
Cost	1484

\[\begin{array}{l} t_0 := \frac{a \cdot b}{x-scale \cdot y-scale}\\ t_1 := -4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{if}\;a \leq 2.81 \cdot 10^{-269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 10^{+23}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+253}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{\frac{a}{y-scale}}{\frac{x-scale}{b} \cdot \frac{y-scale}{a}}\right)\\ \end{array} \]

Alternative 2
Error	8.8
Cost	1353

\[\begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+207} \lor \neg \left(a \leq 2.1 \cdot 10^{+251}\right):\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \left(\frac{a}{y-scale} \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{\frac{a}{x-scale}}{x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 3
Error	6.1
Cost	1353

\[\begin{array}{l} t_0 := \frac{a \cdot b}{x-scale \cdot y-scale}\\ \mathbf{if}\;a \leq 10^{-217} \lor \neg \left(a \leq 1.5 \cdot 10^{-37}\right):\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{b}{x-scale} \cdot \left(\frac{a}{y-scale} \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	8.6
Cost	1088

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

Alternative 5
Error	30.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023054 
(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))))