Simplification of discriminant from scale-rotated-ellipse

Average Error: 40.7 → 24.3

Time: 1.2min

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := {\left(\frac{b}{y-scale}\right)}^{2}\\ t_2 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_3 := \cos t_2\\ t_4 := \sin t_2\\ t_5 := \pi \cdot \frac{angle}{180}\\ t_6 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_5\right) \cdot \cos t_5}{x-scale}}{y-scale}\\ t_7 := \frac{{t_3}^{2}}{x-scale} \cdot \left(\frac{{t_4}^{2}}{x-scale} \cdot t_1\right)\\ t_8 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_9 := \cos t_8\\ t_10 := \sin t_8\\ \mathbf{if}\;x-scale \leq -5 \cdot 10^{-114}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-8, t_7, t_1 \cdot \left(\left(\frac{{t_4}^{4}}{x-scale \cdot x-scale} + \frac{{t_3}^{4}}{x-scale \cdot x-scale}\right) \cdot -4\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -5.8 \cdot 10^{-292}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-8, t_7, -4 \cdot \left(\frac{{\cos t_0}^{4} + {\sin t_0}^{4}}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 10^{-160}:\\ \;\;\;\;t_6 \cdot t_6 + \frac{a}{\frac{y-scale \cdot y-scale}{a}} \cdot \left(-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-8, t_1 \cdot \left(\frac{{t_9}^{2}}{x-scale} \cdot \frac{{t_10}^{2}}{x-scale}\right), t_1 \cdot \left(-4 \cdot \left(\frac{{t_10}^{4}}{x-scale \cdot x-scale} + \frac{{t_9}^{4}}{x-scale \cdot x-scale}\right)\right)\right)\right)\\ \end{array} \]

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 (* angle (* PI 0.005555555555555556)))
        (t_1 (pow (/ b y-scale) 2.0))
        (t_2 (* (* PI angle) 0.005555555555555556))
        (t_3 (cos t_2))
        (t_4 (sin t_2))
        (t_5 (* PI (/ angle 180.0)))
        (t_6
         (/
          (/
           (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_5)) (cos t_5))
           x-scale)
          y-scale))
        (t_7 (* (/ (pow t_3 2.0) x-scale) (* (/ (pow t_4 2.0) x-scale) t_1)))
        (t_8 (* PI (* angle 0.005555555555555556)))
        (t_9 (cos t_8))
        (t_10 (sin t_8)))
   (if (<= x-scale -5e-114)
     (*
      a
      (*
       a
       (fma
        -8.0
        t_7
        (*
         t_1
         (*
          (+
           (/ (pow t_4 4.0) (* x-scale x-scale))
           (/ (pow t_3 4.0) (* x-scale x-scale)))
          -4.0)))))
     (if (<= x-scale -5.8e-292)
       (*
        a
        (*
         a
         (fma
          -8.0
          t_7
          (*
           -4.0
           (*
            (/
             (+ (pow (cos t_0) 4.0) (pow (sin t_0) 4.0))
             (* (* x-scale y-scale) (* x-scale y-scale)))
            (* b b))))))
       (if (<= x-scale 1e-160)
         (+
          (* t_6 t_6)
          (*
           (/ a (/ (* y-scale y-scale) a))
           (* -4.0 (* (/ b x-scale) (/ b x-scale)))))
         (*
          a
          (*
           a
           (fma
            -8.0
            (* t_1 (* (/ (pow t_9 2.0) x-scale) (/ (pow t_10 2.0) x-scale)))
            (*
             t_1
             (*
              -4.0
              (+
               (/ (pow t_10 4.0) (* x-scale x-scale))
               (/ (pow t_9 4.0) (* x-scale x-scale)))))))))))))

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 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_1 = pow((b / y_45_scale), 2.0);
	double t_2 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_3 = cos(t_2);
	double t_4 = sin(t_2);
	double t_5 = ((double) M_PI) * (angle / 180.0);
	double t_6 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_5)) * cos(t_5)) / x_45_scale) / y_45_scale;
	double t_7 = (pow(t_3, 2.0) / x_45_scale) * ((pow(t_4, 2.0) / x_45_scale) * t_1);
	double t_8 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_9 = cos(t_8);
	double t_10 = sin(t_8);
	double tmp;
	if (x_45_scale <= -5e-114) {
		tmp = a * (a * fma(-8.0, t_7, (t_1 * (((pow(t_4, 4.0) / (x_45_scale * x_45_scale)) + (pow(t_3, 4.0) / (x_45_scale * x_45_scale))) * -4.0))));
	} else if (x_45_scale <= -5.8e-292) {
		tmp = a * (a * fma(-8.0, t_7, (-4.0 * (((pow(cos(t_0), 4.0) + pow(sin(t_0), 4.0)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * (b * b)))));
	} else if (x_45_scale <= 1e-160) {
		tmp = (t_6 * t_6) + ((a / ((y_45_scale * y_45_scale) / a)) * (-4.0 * ((b / x_45_scale) * (b / x_45_scale))));
	} else {
		tmp = a * (a * fma(-8.0, (t_1 * ((pow(t_9, 2.0) / x_45_scale) * (pow(t_10, 2.0) / x_45_scale))), (t_1 * (-4.0 * ((pow(t_10, 4.0) / (x_45_scale * x_45_scale)) + (pow(t_9, 4.0) / (x_45_scale * x_45_scale)))))));
	}
	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(angle * Float64(pi * 0.005555555555555556))
	t_1 = Float64(b / y_45_scale) ^ 2.0
	t_2 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_3 = cos(t_2)
	t_4 = sin(t_2)
	t_5 = Float64(pi * Float64(angle / 180.0))
	t_6 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_5)) * cos(t_5)) / x_45_scale) / y_45_scale)
	t_7 = Float64(Float64((t_3 ^ 2.0) / x_45_scale) * Float64(Float64((t_4 ^ 2.0) / x_45_scale) * t_1))
	t_8 = Float64(pi * Float64(angle * 0.005555555555555556))
	t_9 = cos(t_8)
	t_10 = sin(t_8)
	tmp = 0.0
	if (x_45_scale <= -5e-114)
		tmp = Float64(a * Float64(a * fma(-8.0, t_7, Float64(t_1 * Float64(Float64(Float64((t_4 ^ 4.0) / Float64(x_45_scale * x_45_scale)) + Float64((t_3 ^ 4.0) / Float64(x_45_scale * x_45_scale))) * -4.0)))));
	elseif (x_45_scale <= -5.8e-292)
		tmp = Float64(a * Float64(a * fma(-8.0, t_7, Float64(-4.0 * Float64(Float64(Float64((cos(t_0) ^ 4.0) + (sin(t_0) ^ 4.0)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * Float64(b * b))))));
	elseif (x_45_scale <= 1e-160)
		tmp = Float64(Float64(t_6 * t_6) + Float64(Float64(a / Float64(Float64(y_45_scale * y_45_scale) / a)) * Float64(-4.0 * Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale)))));
	else
		tmp = Float64(a * Float64(a * fma(-8.0, Float64(t_1 * Float64(Float64((t_9 ^ 2.0) / x_45_scale) * Float64((t_10 ^ 2.0) / x_45_scale))), Float64(t_1 * Float64(-4.0 * Float64(Float64((t_10 ^ 4.0) / Float64(x_45_scale * x_45_scale)) + Float64((t_9 ^ 4.0) / Float64(x_45_scale * x_45_scale))))))));
	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[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(b / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$5 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$5], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$5], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[Power[t$95$3, 2.0], $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[(N[Power[t$95$4, 2.0], $MachinePrecision] / x$45$scale), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[Cos[t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[Sin[t$95$8], $MachinePrecision]}, If[LessEqual[x$45$scale, -5e-114], N[(a * N[(a * N[(-8.0 * t$95$7 + N[(t$95$1 * N[(N[(N[(N[Power[t$95$4, 4.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$3, 4.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -5.8e-292], N[(a * N[(a * N[(-8.0 * t$95$7 + N[(-4.0 * N[(N[(N[(N[Power[N[Cos[t$95$0], $MachinePrecision], 4.0], $MachinePrecision] + N[Power[N[Sin[t$95$0], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 1e-160], N[(N[(t$95$6 * t$95$6), $MachinePrecision] + N[(N[(a / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(-8.0 * N[(t$95$1 * N[(N[(N[Power[t$95$9, 2.0], $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[Power[t$95$10, 2.0], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(-4.0 * N[(N[(N[Power[t$95$10, 4.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$9, 4.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

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 4 regimes

2. if x-scale < -4.99999999999999989e-114

   1. Initial program 37.7

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

   2. Taylor expanded in a around inf 33.4

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

   3. Simplified 24.9

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

   4. Applied egg-rr 19.6

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

   if -4.99999999999999989e-114 < x-scale < -5.79999999999999985e-292

   1. Initial program 50.8

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

   2. Taylor expanded in a around inf 57.6

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

   3. Simplified 55.8

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

   4. Applied egg-rr 55.1

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

   5. Taylor expanded in x-scale around 0 57.1

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

   6. Simplified 40.7

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

   if -5.79999999999999985e-292 < x-scale < 9.9999999999999999e-161

   1. Initial program 52.3

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

   2. Taylor expanded in angle around 0 63.8

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

   3. Simplified 49.3

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

   if 9.9999999999999999e-161 < x-scale

   1. Initial program 39.0

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

   2. Taylor expanded in a around inf 34.5

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

   3. Simplified 25.8

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

   4. Applied egg-rr 20.2

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

   5. Applied egg-rr 20.2

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

4. Final simplification 24.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -5 \cdot 10^{-114}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-8, \frac{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \left(\frac{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot {\left(\frac{b}{y-scale}\right)}^{2}\right), {\left(\frac{b}{y-scale}\right)}^{2} \cdot \left(\left(\frac{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}{x-scale \cdot x-scale} + \frac{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}{x-scale \cdot x-scale}\right) \cdot -4\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -5.8 \cdot 10^{-292}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-8, \frac{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot \left(\frac{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale} \cdot {\left(\frac{b}{y-scale}\right)}^{2}\right), -4 \cdot \left(\frac{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4} + {\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{4}}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 10^{-160}:\\ \;\;\;\;\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)}{x-scale}}{y-scale} + \frac{a}{\frac{y-scale \cdot y-scale}{a}} \cdot \left(-4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(-8, {\left(\frac{b}{y-scale}\right)}^{2} \cdot \left(\frac{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}{x-scale} \cdot \frac{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}{x-scale}\right), {\left(\frac{b}{y-scale}\right)}^{2} \cdot \left(-4 \cdot \left(\frac{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}{x-scale \cdot x-scale} + \frac{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}{x-scale \cdot x-scale}\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022178 
(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))))