a from scale-rotated-ellipse

Average Error: 63.4 → 47.2

Time: 2.1min

Precision: binary64

Cost: 52804

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;y-scale \leq -5.3:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2 \cdot \left({\left(\cos t_0 \cdot b\right)}^{2} + {\left(a \cdot \sin t_0\right)}^{2}\right)} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\\ \mathbf{elif}\;y-scale \leq -2.1 \cdot 10^{-295}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot x-scale\right)\right)\\ \mathbf{elif}\;y-scale \leq 8.5 \cdot 10^{+31}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|b \cdot \sqrt{8}\right|\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (*
      (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0)))
      (* (* b a) (* b (- a))))
     (+
      (+
       (/
        (/
         (+
          (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))
      (sqrt
       (+
        (pow
         (-
          (/
           (/
            (+
             (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))
         2.0)
        (pow
         (/
          (/
           (*
            (*
             (* 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)))))))
  (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))))
   (if (<= y-scale -5.3)
     (*
      -0.25
      (*
       (sqrt (* 2.0 (+ (pow (* (cos t_0) b) 2.0) (pow (* a (sin t_0)) 2.0))))
       (* y-scale (sqrt 8.0))))
     (if (<= y-scale -2.1e-295)
       (*
        0.25
        (*
         (sqrt 2.0)
         (*
          (* (sqrt 8.0) (* a (cos (* angle (* PI 0.005555555555555556)))))
          x-scale)))
       (if (<= y-scale 8.5e+31)
         (* (* -0.25 x-scale) (* (sqrt 8.0) (* a (sqrt 2.0))))
         (* (* y-scale 0.25) (* (sqrt 2.0) (fabs (* b (sqrt 8.0))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((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)) + sqrt((pow(((((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)), 2.0) + pow((((((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)))))) / ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale <= -5.3) {
		tmp = -0.25 * (sqrt((2.0 * (pow((cos(t_0) * b), 2.0) + pow((a * sin(t_0)), 2.0)))) * (y_45_scale * sqrt(8.0)));
	} else if (y_45_scale <= -2.1e-295) {
		tmp = 0.25 * (sqrt(2.0) * ((sqrt(8.0) * (a * cos((angle * (((double) M_PI) * 0.005555555555555556))))) * x_45_scale));
	} else if (y_45_scale <= 8.5e+31) {
		tmp = (-0.25 * x_45_scale) * (sqrt(8.0) * (a * sqrt(2.0)));
	} else {
		tmp = (y_45_scale * 0.25) * (sqrt(2.0) * fabs((b * sqrt(8.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -Math.sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / Math.pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((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)) + Math.sqrt((Math.pow(((((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)), 2.0) + Math.pow((((((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)))))) / ((4.0 * ((b * a) * (b * -a))) / Math.pow((x_45_scale * y_45_scale), 2.0));
}

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale <= -5.3) {
		tmp = -0.25 * (Math.sqrt((2.0 * (Math.pow((Math.cos(t_0) * b), 2.0) + Math.pow((a * Math.sin(t_0)), 2.0)))) * (y_45_scale * Math.sqrt(8.0)));
	} else if (y_45_scale <= -2.1e-295) {
		tmp = 0.25 * (Math.sqrt(2.0) * ((Math.sqrt(8.0) * (a * Math.cos((angle * (Math.PI * 0.005555555555555556))))) * x_45_scale));
	} else if (y_45_scale <= 8.5e+31) {
		tmp = (-0.25 * x_45_scale) * (Math.sqrt(8.0) * (a * Math.sqrt(2.0)));
	} else {
		tmp = (y_45_scale * 0.25) * (Math.sqrt(2.0) * Math.abs((b * Math.sqrt(8.0))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return -math.sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / math.pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((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)) + math.sqrt((math.pow(((((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)), 2.0) + math.pow((((((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)))))) / ((4.0 * ((b * a) * (b * -a))) / math.pow((x_45_scale * y_45_scale), 2.0))

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pi * (0.005555555555555556 * angle)
	tmp = 0
	if y_45_scale <= -5.3:
		tmp = -0.25 * (math.sqrt((2.0 * (math.pow((math.cos(t_0) * b), 2.0) + math.pow((a * math.sin(t_0)), 2.0)))) * (y_45_scale * math.sqrt(8.0)))
	elif y_45_scale <= -2.1e-295:
		tmp = 0.25 * (math.sqrt(2.0) * ((math.sqrt(8.0) * (a * math.cos((angle * (math.pi * 0.005555555555555556))))) * x_45_scale))
	elif y_45_scale <= 8.5e+31:
		tmp = (-0.25 * x_45_scale) * (math.sqrt(8.0) * (a * math.sqrt(2.0)))
	else:
		tmp = (y_45_scale * 0.25) * (math.sqrt(2.0) * math.fabs((b * math.sqrt(8.0))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0))) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * Float64(Float64(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)) + sqrt(Float64((Float64(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)) ^ 2.0) + (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) ^ 2.0))))))) / Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0)))
end

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (y_45_scale <= -5.3)
		tmp = Float64(-0.25 * Float64(sqrt(Float64(2.0 * Float64((Float64(cos(t_0) * b) ^ 2.0) + (Float64(a * sin(t_0)) ^ 2.0)))) * Float64(y_45_scale * sqrt(8.0))));
	elseif (y_45_scale <= -2.1e-295)
		tmp = Float64(0.25 * Float64(sqrt(2.0) * Float64(Float64(sqrt(8.0) * Float64(a * cos(Float64(angle * Float64(pi * 0.005555555555555556))))) * x_45_scale)));
	elseif (y_45_scale <= 8.5e+31)
		tmp = Float64(Float64(-0.25 * x_45_scale) * Float64(sqrt(8.0) * Float64(a * sqrt(2.0))));
	else
		tmp = Float64(Float64(y_45_scale * 0.25) * Float64(sqrt(2.0) * abs(Float64(b * sqrt(8.0)))));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = -sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / ((x_45_scale * y_45_scale) ^ 2.0))) * ((b * a) * (b * -a))) * (((((((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)) + sqrt(((((((((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)) ^ 2.0) + ((((((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)))))) / ((4.0 * ((b * a) * (b * -a))) / ((x_45_scale * y_45_scale) ^ 2.0));
end

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = pi * (0.005555555555555556 * angle);
	tmp = 0.0;
	if (y_45_scale <= -5.3)
		tmp = -0.25 * (sqrt((2.0 * (((cos(t_0) * b) ^ 2.0) + ((a * sin(t_0)) ^ 2.0)))) * (y_45_scale * sqrt(8.0)));
	elseif (y_45_scale <= -2.1e-295)
		tmp = 0.25 * (sqrt(2.0) * ((sqrt(8.0) * (a * cos((angle * (pi * 0.005555555555555556))))) * x_45_scale));
	elseif (y_45_scale <= 8.5e+31)
		tmp = (-0.25 * x_45_scale) * (sqrt(8.0) * (a * sqrt(2.0)));
	else
		tmp = (y_45_scale * 0.25) * (sqrt(2.0) * abs((b * sqrt(8.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[((-N[Sqrt[N[(N[(N[(2.0 * N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(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] + 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] + N[Sqrt[N[(N[Power[N[(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] - 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], 2.0], $MachinePrecision] + N[Power[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], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -5.3], N[(-0.25 * N[(N[Sqrt[N[(2.0 * N[(N[Power[N[(N[Cos[t$95$0], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, -2.1e-295], N[(0.25 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[(a * N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 8.5e+31], N[(N[(-0.25 * x$45$scale), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(a * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$45$scale * 0.25), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Abs[N[(b * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y-scale < -5.29999999999999982

   1. Initial program 63.3

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

   2. Simplified 63.3

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

      [Start] 63.3

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

   3. Taylor expanded in y-scale around -inf 59.6

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right)} \]

   4. Taylor expanded in x-scale around 0 43.6

      \[\leadsto -0.25 \cdot \color{blue}{\left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]

   5. Simplified 40.4

      \[\leadsto -0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \left({\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\right)} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \]
      Proof

      [Start] 43.6	\[ -0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      *-commutative [=>] 43.6	\[ -0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \]

   if -5.29999999999999982 < y-scale < -2.09999999999999993e-295

   1. Initial program 63.5

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

   2. Simplified 63.5

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

      [Start] 63.5

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

   3. Taylor expanded in x-scale around -inf 61.8

      \[\leadsto \color{blue}{\left(-0.25 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   4. Simplified 61.7

      \[\leadsto \color{blue}{\left(\left(-0.25 \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right) \cdot \sqrt{2 \cdot \left(\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}}\right)}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      Proof

      [Start] 61.8	\[ \left(-0.25 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      associate-*r* [=>] 61.8	\[ \color{blue}{\left(\left(-0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot y-scale}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      *-commutative [<=] 61.8	\[ \left(\left(-0.25 \cdot \frac{\sqrt{8}}{\color{blue}{y-scale \cdot x-scale}}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      distribute-lft-out [=>] 61.8	\[ \left(\left(-0.25 \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}}\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      +-commutative [<=] 61.8	\[ \left(\left(-0.25 \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right) \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}}\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   5. Taylor expanded in a around -inf 51.5

      \[\leadsto \color{blue}{0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)} \]

   6. Simplified 51.6

      \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{2} \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot x-scale\right)\right)} \]
      Proof

      [Start] 51.5	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right) \]
      *-commutative [=>] 51.5	\[ 0.25 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right) \cdot x-scale\right)} \]
      associate-*l* [=>] 51.5	\[ 0.25 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right) \cdot x-scale\right)\right)} \]
      associate-*r* [=>] 51.5	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \sqrt{8}\right)} \cdot x-scale\right)\right) \]
      *-commutative [=>] 51.5	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\sqrt{8} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot x-scale\right)\right) \]
      *-commutative [=>] 51.5	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \cdot x-scale\right)\right) \]
      associate-*l* [=>] 51.6	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot x-scale\right)\right) \]

   if -2.09999999999999993e-295 < y-scale < 8.49999999999999947e31

   1. Initial program 63.4

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

   2. Simplified 63.5

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

      [Start] 63.4

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

   3. Taylor expanded in x-scale around -inf 61.8

      \[\leadsto \color{blue}{\left(-0.25 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   4. Simplified 61.8

      \[\leadsto \color{blue}{\left(\left(-0.25 \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right) \cdot \sqrt{2 \cdot \left(\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}}\right)}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      Proof

      [Start] 61.8	\[ \left(-0.25 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      associate-*r* [=>] 61.8	\[ \color{blue}{\left(\left(-0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot y-scale}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      *-commutative [<=] 61.8	\[ \left(\left(-0.25 \cdot \frac{\sqrt{8}}{\color{blue}{y-scale \cdot x-scale}}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      distribute-lft-out [=>] 61.8	\[ \left(\left(-0.25 \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}}\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
      +-commutative [<=] 61.8	\[ \left(\left(-0.25 \cdot \frac{\sqrt{8}}{y-scale \cdot x-scale}\right) \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}}\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   5. Taylor expanded in angle around 0 51.3

      \[\leadsto \color{blue}{-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)\right)} \]

   6. Simplified 51.3

      \[\leadsto \color{blue}{\left(-0.25 \cdot x-scale\right) \cdot \left(\left(\sqrt{2} \cdot a\right) \cdot \sqrt{8}\right)} \]
      Proof

      [Start] 51.3	\[ -0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 51.3	\[ \color{blue}{\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)} \]
      associate-*r* [=>] 51.3	\[ \left(-0.25 \cdot x-scale\right) \cdot \color{blue}{\left(\left(\sqrt{2} \cdot a\right) \cdot \sqrt{8}\right)} \]

   if 8.49999999999999947e31 < y-scale

   1. Initial program 63.3

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

   2. Simplified 63.5

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

      [Start] 63.3

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

   3. Taylor expanded in angle around 0 51.9

      \[\leadsto \color{blue}{0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right)} \]

   4. Simplified 51.9

      \[\leadsto \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\left(\sqrt{8} \cdot b\right) \cdot \sqrt{2}\right)} \]
      Proof

      [Start] 51.9	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 51.9	\[ \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 51.9	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]
      *-commutative [=>] 51.9	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left(\sqrt{8} \cdot b\right)} \cdot \sqrt{2}\right) \]

   5. Applied egg-rr 46.2

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\sqrt{8 \cdot \left(b \cdot b\right)}} \cdot \sqrt{2}\right) \]

   6. Simplified 40.0

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left|b \cdot \sqrt{8}\right|} \cdot \sqrt{2}\right) \]
      Proof

      [Start] 46.2	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{8 \cdot \left(b \cdot b\right)} \cdot \sqrt{2}\right) \]
      rem-square-sqrt [<=] 46.4	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{\color{blue}{\left(\sqrt{8} \cdot \sqrt{8}\right)} \cdot \left(b \cdot b\right)} \cdot \sqrt{2}\right) \]
      swap-sqr [<=] 46.3	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{\color{blue}{\left(\sqrt{8} \cdot b\right) \cdot \left(\sqrt{8} \cdot b\right)}} \cdot \sqrt{2}\right) \]
      rem-sqrt-square [=>] 40.0	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left|\sqrt{8} \cdot b\right|} \cdot \sqrt{2}\right) \]
      *-commutative [=>] 40.0	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\left|\color{blue}{b \cdot \sqrt{8}}\right| \cdot \sqrt{2}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 47.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -5.3:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2 \cdot \left({\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\right)} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\\ \mathbf{elif}\;y-scale \leq -2.1 \cdot 10^{-295}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot x-scale\right)\right)\\ \mathbf{elif}\;y-scale \leq 8.5 \cdot 10^{+31}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|b \cdot \sqrt{8}\right|\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	51.6
Cost	27156

\[\begin{array}{l} t_0 := \sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\\ t_1 := t_0 \cdot \left(0.25 \cdot x-scale\right)\\ \mathbf{if}\;a \leq -2.1 \cdot 10^{+243}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;a \leq -1.9 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{+41}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot t_0\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{-67}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot \sqrt{\left(a \cdot a\right) \cdot 16}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-178}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	51.7
Cost	27156

\[\begin{array}{l} t_0 := x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ t_1 := \sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\\ t_2 := t_1 \cdot \left(0.25 \cdot x-scale\right)\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{+243}:\\ \;\;\;\;-0.25 \cdot t_0\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{+162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{+75}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot t_1\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{-73}:\\ \;\;\;\;0.25 \cdot t_0\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-178}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	51.7
Cost	27156

\[\begin{array}{l} t_0 := \sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\\ t_1 := t_0 \cdot \left(0.25 \cdot x-scale\right)\\ \mathbf{if}\;a \leq -5.2 \cdot 10^{+240}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{+161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{+74}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot t_0\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-73}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(\left(\sqrt{8} \cdot \left(a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot x-scale\right)\right)\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-178}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	51.6
Cost	26628

\[\begin{array}{l} t_0 := \sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\\ t_1 := t_0 \cdot \left(0.25 \cdot x-scale\right)\\ \mathbf{if}\;a \leq -4 \cdot 10^{+243}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{+41}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot t_0\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-69}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot \sqrt{\left(a \cdot a\right) \cdot 16}\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-200}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	51.5
Cost	14036

\[\begin{array}{l} t_0 := \sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\\ t_1 := t_0 \cdot \left(0.25 \cdot x-scale\right)\\ \mathbf{if}\;a \leq -1.4 \cdot 10^{+243}:\\ \;\;\;\;-0.25 \cdot \left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right)\\ \mathbf{elif}\;a \leq -1.76 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{+41}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot t_0\\ \mathbf{elif}\;a \leq -2.3 \cdot 10^{-70}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot \sqrt{\left(a \cdot a\right) \cdot 16}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-201}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	51.2
Cost	13904

\[\begin{array}{l} t_0 := -0.25 \cdot \left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right)\\ \mathbf{if}\;a \leq -6.2 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -4.4 \cdot 10^{-67}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot \sqrt{\left(a \cdot a\right) \cdot 16}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-232}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	51.2
Cost	13904

\[\begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+41}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-66}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot \sqrt{\left(a \cdot a\right) \cdot 16}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-233}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;a \leq 0.00045:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	53.7
Cost	7240

\[\begin{array}{l} t_0 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{if}\;a \leq -6.2 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;\left(-0.25 \cdot x-scale\right) \cdot \sqrt{\left(a \cdot a\right) \cdot 16}\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-233}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 9
Error	54.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;angle \leq -8.2 \cdot 10^{-243} \lor \neg \left(angle \leq 3.6 \cdot 10^{-122}\right):\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 10
Error	54.2
Cost	448

\[\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right) \]

Reproduce

herbie shell --seed 2023054 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (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)) (sqrt (+ (pow (- (/ (/ (+ (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)) 2.0) (pow (/ (/ (* (* (* 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))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))