Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.2 → 14.0

Time: 1.9min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a \cdot a}{-0.25}\\ t_1 := -4 \cdot \left(a \cdot \left(b \cdot \frac{a \cdot b}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ t_2 := {\left(\frac{b}{x-scale \cdot y-scale}\right)}^{2}\\ \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\frac{t_0}{\frac{1}{t_2}}\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{t_2}{\frac{1}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 a) -0.25))
        (t_1
         (*
          -4.0
          (* a (* b (/ (* a b) (* (* x-scale y-scale) (* x-scale y-scale)))))))
        (t_2 (pow (/ b (* x-scale y-scale)) 2.0)))
   (if (<= a -2e+154)
     t_1
     (if (<= a -1e-90)
       (/ t_0 (/ 1.0 t_2))
       (if (<= a 1.95e-172)
         t_1
         (if (<= a 6.8e+153) (/ t_2 (/ 1.0 t_0)) t_1))))))

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 * a) / -0.25;
	double t_1 = -4.0 * (a * (b * ((a * b) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	double t_2 = pow((b / (x_45_scale * y_45_scale)), 2.0);
	double tmp;
	if (a <= -2e+154) {
		tmp = t_1;
	} else if (a <= -1e-90) {
		tmp = t_0 / (1.0 / t_2);
	} else if (a <= 1.95e-172) {
		tmp = t_1;
	} else if (a <= 6.8e+153) {
		tmp = t_2 / (1.0 / t_0);
	} else {
		tmp = t_1;
	}
	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 * a) / -0.25;
	double t_1 = -4.0 * (a * (b * ((a * b) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	double t_2 = Math.pow((b / (x_45_scale * y_45_scale)), 2.0);
	double tmp;
	if (a <= -2e+154) {
		tmp = t_1;
	} else if (a <= -1e-90) {
		tmp = t_0 / (1.0 / t_2);
	} else if (a <= 1.95e-172) {
		tmp = t_1;
	} else if (a <= 6.8e+153) {
		tmp = t_2 / (1.0 / t_0);
	} else {
		tmp = t_1;
	}
	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 * a) / -0.25
	t_1 = -4.0 * (a * (b * ((a * b) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))))
	t_2 = math.pow((b / (x_45_scale * y_45_scale)), 2.0)
	tmp = 0
	if a <= -2e+154:
		tmp = t_1
	elif a <= -1e-90:
		tmp = t_0 / (1.0 / t_2)
	elif a <= 1.95e-172:
		tmp = t_1
	elif a <= 6.8e+153:
		tmp = t_2 / (1.0 / t_0)
	else:
		tmp = t_1
	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 * a) / -0.25)
	t_1 = Float64(-4.0 * Float64(a * Float64(b * Float64(Float64(a * b) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))))))
	t_2 = Float64(b / Float64(x_45_scale * y_45_scale)) ^ 2.0
	tmp = 0.0
	if (a <= -2e+154)
		tmp = t_1;
	elseif (a <= -1e-90)
		tmp = Float64(t_0 / Float64(1.0 / t_2));
	elseif (a <= 1.95e-172)
		tmp = t_1;
	elseif (a <= 6.8e+153)
		tmp = Float64(t_2 / Float64(1.0 / t_0));
	else
		tmp = t_1;
	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 * a) / -0.25;
	t_1 = -4.0 * (a * (b * ((a * b) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale)))));
	t_2 = (b / (x_45_scale * y_45_scale)) ^ 2.0;
	tmp = 0.0;
	if (a <= -2e+154)
		tmp = t_1;
	elseif (a <= -1e-90)
		tmp = t_0 / (1.0 / t_2);
	elseif (a <= 1.95e-172)
		tmp = t_1;
	elseif (a <= 6.8e+153)
		tmp = t_2 / (1.0 / t_0);
	else
		tmp = t_1;
	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 * a), $MachinePrecision] / -0.25), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(a * N[(b * N[(N[(a * b), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[a, -2e+154], t$95$1, If[LessEqual[a, -1e-90], N[(t$95$0 / N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.95e-172], t$95$1, If[LessEqual[a, 6.8e+153], N[(t$95$2 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Bits error versus x-scale

Bits error versus y-scale

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000007e154 or -9.99999999999999995e-91 < a < 1.94999999999999986e-172 or 6.7999999999999995e153 < a

   1. Initial program 42.3

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

   2. Taylor expanded in y-scale around 0 43.7

      \[\leadsto \color{blue}{-1 \cdot \frac{4 \cdot \frac{{a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}{{x-scale}^{2}} + \left(8 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{{x-scale}^{2}} + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2}}\right)}{{y-scale}^{2}}} \]

   3. Simplified 44.5

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, \left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{x-scale \cdot x-scale}\right), \mathsf{fma}\left(8, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right), 4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot \left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)\right)}{-y-scale \cdot y-scale}} \]

   4. Taylor expanded in angle around 0 44.1

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

   5. Simplified 40.5

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

   6. Taylor expanded in b around 0 44.1

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

   7. Simplified 18.2

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

   if -2.00000000000000007e154 < a < -9.99999999999999995e-91

   1. Initial program 40.9

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

   2. Taylor expanded in y-scale around 0 33.8

      \[\leadsto \color{blue}{-1 \cdot \frac{4 \cdot \frac{{a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}{{x-scale}^{2}} + \left(8 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{{x-scale}^{2}} + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2}}\right)}{{y-scale}^{2}}} \]

   3. Simplified 32.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, \left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{x-scale \cdot x-scale}\right), \mathsf{fma}\left(8, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right), 4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot \left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)\right)}{-y-scale \cdot y-scale}} \]

   4. Taylor expanded in angle around 0 33.5

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

   5. Simplified 29.2

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

   6. Applied egg-rr 7.8

      \[\leadsto \color{blue}{\frac{\frac{a \cdot a}{-0.25}}{\frac{1}{{\left(\frac{b}{x-scale \cdot y-scale}\right)}^{2}}}} \]

   if 1.94999999999999986e-172 < a < 6.7999999999999995e153

   1. Initial program 39.3

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

   2. Taylor expanded in y-scale around 0 35.1

      \[\leadsto \color{blue}{-1 \cdot \frac{4 \cdot \frac{{a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right)}{{x-scale}^{2}} + \left(8 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}{{x-scale}^{2}} + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2}}\right)}{{y-scale}^{2}}} \]

   3. Simplified 34.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, \left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{x-scale \cdot x-scale}\right), \mathsf{fma}\left(8, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right), 4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot \left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{x-scale \cdot x-scale}\right)\right)\right)\right)}{-y-scale \cdot y-scale}} \]

   4. Taylor expanded in angle around 0 35.3

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

   5. Simplified 30.7

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

   6. Applied egg-rr 10.2

      \[\leadsto \color{blue}{\frac{{\left(\frac{b}{x-scale \cdot y-scale}\right)}^{2}}{\frac{1}{\frac{a \cdot a}{-0.25}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-4 \cdot \left(a \cdot \left(b \cdot \frac{a \cdot b}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{a \cdot a}{-0.25}}{\frac{1}{{\left(\frac{b}{x-scale \cdot y-scale}\right)}^{2}}}\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-172}:\\ \;\;\;\;-4 \cdot \left(a \cdot \left(b \cdot \frac{a \cdot b}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{{\left(\frac{b}{x-scale \cdot y-scale}\right)}^{2}}{\frac{1}{\frac{a \cdot a}{-0.25}}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a \cdot \left(b \cdot \frac{a \cdot b}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(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))))