Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.0 → 26.0

Time: 1.4min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t_0\\ t_2 := \sin t_0\\ t_3 := 8 \cdot \left({x-scale}^{-2} \cdot {\left(\left(a \cdot t_1\right) \cdot \left(b \cdot t_2\right)\right)}^{2}\right)\\ t_4 := {\left(a \cdot b\right)}^{2}\\ t_5 := \left(t_4 \cdot {t_2}^{4}\right) \cdot {x-scale}^{-2}\\ t_6 := t_4 \cdot {t_1}^{4}\\ \mathbf{if}\;x-scale \leq -1.051336110722494 \cdot 10^{+131}:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq -9.808554852601608 \cdot 10^{-151}:\\ \;\;\;\;-{\left({\left(\sqrt{\frac{y-scale \cdot y-scale}{\mathsf{fma}\left(4, t_5, \mathsf{fma}\left(4, {x-scale}^{-2} \cdot t_6, t_3\right)\right)}}\right)}^{2}\right)}^{-1}\\ \mathbf{elif}\;x-scale \leq 7.474864398681438 \cdot 10^{-155}:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq 7.39453938945273 \cdot 10^{+110}:\\ \;\;\;\;-{\left(\frac{y-scale \cdot y-scale}{\mathsf{fma}\left(4, t_5, \mathsf{fma}\left(4, \frac{t_6}{x-scale \cdot x-scale}, t_3\right)\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

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 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3 (* 8.0 (* (pow x-scale -2.0) (pow (* (* a t_1) (* b t_2)) 2.0))))
        (t_4 (pow (* a b) 2.0))
        (t_5 (* (* t_4 (pow t_2 4.0)) (pow x-scale -2.0)))
        (t_6 (* t_4 (pow t_1 4.0))))
   (if (<= x-scale -1.051336110722494e+131)
     0.0
     (if (<= x-scale -9.808554852601608e-151)
       (-
        (pow
         (pow
          (sqrt
           (/
            (* y-scale y-scale)
            (fma 4.0 t_5 (fma 4.0 (* (pow x-scale -2.0) t_6) t_3))))
          2.0)
         -1.0))
       (if (<= x-scale 7.474864398681438e-155)
         0.0
         (if (<= x-scale 7.39453938945273e+110)
           (-
            (pow
             (/
              (* y-scale y-scale)
              (fma 4.0 t_5 (fma 4.0 (/ t_6 (* x-scale x-scale)) t_3)))
             -1.0))
           0.0))))))

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 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = 8.0 * (pow(x_45_scale, -2.0) * pow(((a * t_1) * (b * t_2)), 2.0));
	double t_4 = pow((a * b), 2.0);
	double t_5 = (t_4 * pow(t_2, 4.0)) * pow(x_45_scale, -2.0);
	double t_6 = t_4 * pow(t_1, 4.0);
	double tmp;
	if (x_45_scale <= -1.051336110722494e+131) {
		tmp = 0.0;
	} else if (x_45_scale <= -9.808554852601608e-151) {
		tmp = -pow(pow(sqrt(((y_45_scale * y_45_scale) / fma(4.0, t_5, fma(4.0, (pow(x_45_scale, -2.0) * t_6), t_3)))), 2.0), -1.0);
	} else if (x_45_scale <= 7.474864398681438e-155) {
		tmp = 0.0;
	} else if (x_45_scale <= 7.39453938945273e+110) {
		tmp = -pow(((y_45_scale * y_45_scale) / fma(4.0, t_5, fma(4.0, (t_6 / (x_45_scale * x_45_scale)), t_3))), -1.0);
	} else {
		tmp = 0.0;
	}
	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(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(8.0 * Float64((x_45_scale ^ -2.0) * (Float64(Float64(a * t_1) * Float64(b * t_2)) ^ 2.0)))
	t_4 = Float64(a * b) ^ 2.0
	t_5 = Float64(Float64(t_4 * (t_2 ^ 4.0)) * (x_45_scale ^ -2.0))
	t_6 = Float64(t_4 * (t_1 ^ 4.0))
	tmp = 0.0
	if (x_45_scale <= -1.051336110722494e+131)
		tmp = 0.0;
	elseif (x_45_scale <= -9.808554852601608e-151)
		tmp = Float64(-((sqrt(Float64(Float64(y_45_scale * y_45_scale) / fma(4.0, t_5, fma(4.0, Float64((x_45_scale ^ -2.0) * t_6), t_3)))) ^ 2.0) ^ -1.0));
	elseif (x_45_scale <= 7.474864398681438e-155)
		tmp = 0.0;
	elseif (x_45_scale <= 7.39453938945273e+110)
		tmp = Float64(-(Float64(Float64(y_45_scale * y_45_scale) / fma(4.0, t_5, fma(4.0, Float64(t_6 / Float64(x_45_scale * x_45_scale)), t_3))) ^ -1.0));
	else
		tmp = 0.0;
	end
	return 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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(8.0 * N[(N[Power[x$45$scale, -2.0], $MachinePrecision] * N[Power[N[(N[(a * t$95$1), $MachinePrecision] * N[(b * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * N[Power[t$95$2, 4.0], $MachinePrecision]), $MachinePrecision] * N[Power[x$45$scale, -2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * N[Power[t$95$1, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.051336110722494e+131], 0.0, If[LessEqual[x$45$scale, -9.808554852601608e-151], (-N[Power[N[Power[N[Sqrt[N[(N[(y$45$scale * y$45$scale), $MachinePrecision] / N[(4.0 * t$95$5 + N[(4.0 * N[(N[Power[x$45$scale, -2.0], $MachinePrecision] * t$95$6), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision]), If[LessEqual[x$45$scale, 7.474864398681438e-155], 0.0, If[LessEqual[x$45$scale, 7.39453938945273e+110], (-N[Power[N[(N[(y$45$scale * y$45$scale), $MachinePrecision] / N[(4.0 * t$95$5 + N[(4.0 * N[(t$95$6 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), 0.0]]]]]]]]]]]

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 x-scale < -1.0513361107224939e131 or -9.80855485260160846e-151 < x-scale < 7.474864398681438e-155 or 7.39453938945273009e110 < x-scale

   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. Taylor expanded in b around 0 43.0

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

   3. Simplified 28.2

      \[\leadsto \color{blue}{0} \]

   if -1.0513361107224939e131 < x-scale < -9.80855485260160846e-151

   1. Initial program 41.2

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

   2. Taylor expanded in y-scale around 0 34.1

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

   3. Simplified 34.1

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

   4. Applied egg-rr 31.1

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

   5. Applied egg-rr 23.4

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

   if 7.474864398681438e-155 < x-scale < 7.39453938945273009e110

   1. Initial program 41.6

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

   2. Taylor expanded in y-scale around 0 33.7

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

   3. Simplified 33.7

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

   4. Applied egg-rr 30.6

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

   5. Applied egg-rr 23.0

      \[\leadsto -{\left(\frac{y-scale \cdot y-scale}{\mathsf{fma}\left(4, \left({\left(a \cdot b\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right) \cdot {x-scale}^{-2}, \mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4} \cdot {\left(a \cdot b\right)}^{2}}{x-scale \cdot x-scale}, 8 \cdot \left(\color{blue}{{\left(\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}} \cdot {x-scale}^{-2}\right)\right)\right)}\right)}^{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.051336110722494 \cdot 10^{+131}:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq -9.808554852601608 \cdot 10^{-151}:\\ \;\;\;\;-{\left({\left(\sqrt{\frac{y-scale \cdot y-scale}{\mathsf{fma}\left(4, \left({\left(a \cdot b\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right) \cdot {x-scale}^{-2}, \mathsf{fma}\left(4, {x-scale}^{-2} \cdot \left({\left(a \cdot b\right)}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right), 8 \cdot \left({x-scale}^{-2} \cdot {\left(\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}\right)\right)\right)}}\right)}^{2}\right)}^{-1}\\ \mathbf{elif}\;x-scale \leq 7.474864398681438 \cdot 10^{-155}:\\ \;\;\;\;0\\ \mathbf{elif}\;x-scale \leq 7.39453938945273 \cdot 10^{+110}:\\ \;\;\;\;-{\left(\frac{y-scale \cdot y-scale}{\mathsf{fma}\left(4, \left({\left(a \cdot b\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}\right) \cdot {x-scale}^{-2}, \mathsf{fma}\left(4, \frac{{\left(a \cdot b\right)}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{x-scale \cdot x-scale}, 8 \cdot \left({x-scale}^{-2} \cdot {\left(\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}\right)\right)\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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))))