a from scale-rotated-ellipse

Average Error: 63.5 → 47.6

Time: 1.8min

Precision: binary64

Cost: 46800

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\\ t_2 := y-scale \cdot \left(t_1 \cdot -0.25\right)\\ t_3 := x-scale \cdot \sqrt{\left(2 \cdot \left({\left(a \cdot \cos t_0\right)}^{2} + {\left(b \cdot \sin t_0\right)}^{2}\right)\right) \cdot 8}\\ \mathbf{if}\;x-scale \leq -9.5 \cdot 10^{+54}:\\ \;\;\;\;t_3 \cdot -0.25\\ \mathbf{elif}\;x-scale \leq -2.45 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq -1.7 \cdot 10^{-252}:\\ \;\;\;\;y-scale \cdot \left(0.25 \cdot t_1\right)\\ \mathbf{elif}\;x-scale \leq 1.15 \cdot 10^{-198}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot t_3\\ \end{array} \]

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 (* angle 0.005555555555555556)))
        (t_1 (* (cos (* angle (* 0.005555555555555556 PI))) (* b 4.0)))
        (t_2 (* y-scale (* t_1 -0.25)))
        (t_3
         (*
          x-scale
          (sqrt
           (*
            (* 2.0 (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)))
            8.0)))))
   (if (<= x-scale -9.5e+54)
     (* t_3 -0.25)
     (if (<= x-scale -2.45e-144)
       t_2
       (if (<= x-scale -1.7e-252)
         (* y-scale (* 0.25 t_1))
         (if (<= x-scale 1.15e-198) t_2 (* 0.25 t_3)))))))

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) * (angle * 0.005555555555555556);
	double t_1 = cos((angle * (0.005555555555555556 * ((double) M_PI)))) * (b * 4.0);
	double t_2 = y_45_scale * (t_1 * -0.25);
	double t_3 = x_45_scale * sqrt(((2.0 * (pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0))) * 8.0));
	double tmp;
	if (x_45_scale <= -9.5e+54) {
		tmp = t_3 * -0.25;
	} else if (x_45_scale <= -2.45e-144) {
		tmp = t_2;
	} else if (x_45_scale <= -1.7e-252) {
		tmp = y_45_scale * (0.25 * t_1);
	} else if (x_45_scale <= 1.15e-198) {
		tmp = t_2;
	} else {
		tmp = 0.25 * t_3;
	}
	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 * (angle * 0.005555555555555556);
	double t_1 = Math.cos((angle * (0.005555555555555556 * Math.PI))) * (b * 4.0);
	double t_2 = y_45_scale * (t_1 * -0.25);
	double t_3 = x_45_scale * Math.sqrt(((2.0 * (Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0))) * 8.0));
	double tmp;
	if (x_45_scale <= -9.5e+54) {
		tmp = t_3 * -0.25;
	} else if (x_45_scale <= -2.45e-144) {
		tmp = t_2;
	} else if (x_45_scale <= -1.7e-252) {
		tmp = y_45_scale * (0.25 * t_1);
	} else if (x_45_scale <= 1.15e-198) {
		tmp = t_2;
	} else {
		tmp = 0.25 * t_3;
	}
	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 * (angle * 0.005555555555555556)
	t_1 = math.cos((angle * (0.005555555555555556 * math.pi))) * (b * 4.0)
	t_2 = y_45_scale * (t_1 * -0.25)
	t_3 = x_45_scale * math.sqrt(((2.0 * (math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0))) * 8.0))
	tmp = 0
	if x_45_scale <= -9.5e+54:
		tmp = t_3 * -0.25
	elif x_45_scale <= -2.45e-144:
		tmp = t_2
	elif x_45_scale <= -1.7e-252:
		tmp = y_45_scale * (0.25 * t_1)
	elif x_45_scale <= 1.15e-198:
		tmp = t_2
	else:
		tmp = 0.25 * t_3
	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(angle * 0.005555555555555556))
	t_1 = Float64(cos(Float64(angle * Float64(0.005555555555555556 * pi))) * Float64(b * 4.0))
	t_2 = Float64(y_45_scale * Float64(t_1 * -0.25))
	t_3 = Float64(x_45_scale * sqrt(Float64(Float64(2.0 * Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))) * 8.0)))
	tmp = 0.0
	if (x_45_scale <= -9.5e+54)
		tmp = Float64(t_3 * -0.25);
	elseif (x_45_scale <= -2.45e-144)
		tmp = t_2;
	elseif (x_45_scale <= -1.7e-252)
		tmp = Float64(y_45_scale * Float64(0.25 * t_1));
	elseif (x_45_scale <= 1.15e-198)
		tmp = t_2;
	else
		tmp = Float64(0.25 * t_3);
	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 * (angle * 0.005555555555555556);
	t_1 = cos((angle * (0.005555555555555556 * pi))) * (b * 4.0);
	t_2 = y_45_scale * (t_1 * -0.25);
	t_3 = x_45_scale * sqrt(((2.0 * (((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0))) * 8.0));
	tmp = 0.0;
	if (x_45_scale <= -9.5e+54)
		tmp = t_3 * -0.25;
	elseif (x_45_scale <= -2.45e-144)
		tmp = t_2;
	elseif (x_45_scale <= -1.7e-252)
		tmp = y_45_scale * (0.25 * t_1);
	elseif (x_45_scale <= 1.15e-198)
		tmp = t_2;
	else
		tmp = 0.25 * t_3;
	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[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y$45$scale * N[(t$95$1 * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x$45$scale * N[Sqrt[N[(N[(2.0 * N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -9.5e+54], N[(t$95$3 * -0.25), $MachinePrecision], If[LessEqual[x$45$scale, -2.45e-144], t$95$2, If[LessEqual[x$45$scale, -1.7e-252], N[(y$45$scale * N[(0.25 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 1.15e-198], t$95$2, N[(0.25 * t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x-scale < -9.4999999999999999e54

   1. Initial program 63.6

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

   2. Simplified 63.4

      \[\leadsto \color{blue}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \frac{b \cdot a}{\sqrt{\left(b \cdot a\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2} + {\left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \frac{\frac{2}{x-scale}}{y-scale}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)}^{2}}\right)\right) \cdot \left(\left(b \cdot \left(-a\right)\right) \cdot \frac{a \cdot \left(\left(\left(-a\right) \cdot \left(b \cdot b\right)\right) \cdot 8\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right)}}\right)}} \]
      Proof

      [Start] 63.6

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

   3. Taylor expanded in x-scale around inf 62.6

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

   4. Simplified 62.3

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

      [Start] 62.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot x-scale}{a \cdot \left(b \cdot \sqrt{8}\right)} \cdot \sqrt{\frac{1}{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right)\right)} \]
      rational.json-simplify-43 [=>] 62.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot x-scale}{\color{blue}{b \cdot \left(\sqrt{8} \cdot a\right)}} \cdot \sqrt{\frac{1}{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right)\right)} \]
      exponential.json-simplify-27 [=>] 62.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot x-scale}{b \cdot \left(\sqrt{8} \cdot a\right)} \cdot \sqrt{\frac{1}{2 \cdot \frac{\color{blue}{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right)\right)} \]
      exponential.json-simplify-27 [=>] 62.3	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot x-scale}{b \cdot \left(\sqrt{8} \cdot a\right)} \cdot \sqrt{\frac{1}{2 \cdot \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{\color{blue}{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}}{{y-scale}^{2}}}}\right)\right)} \]

   5. Taylor expanded in y-scale around -inf 42.2

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

   6. Simplified 39.0

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

      [Start] 42.2	\[ -0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      rational.json-simplify-2 [=>] 42.2	\[ \color{blue}{\left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \cdot -0.25} \]

   if -9.4999999999999999e54 < x-scale < -2.45000000000000005e-144 or -1.7e-252 < x-scale < 1.15000000000000007e-198

   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 63.3

      \[\leadsto \color{blue}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \frac{b \cdot a}{\sqrt{\left(b \cdot a\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2} + {\left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \frac{\frac{2}{x-scale}}{y-scale}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)}^{2}}\right)\right) \cdot \left(\left(b \cdot \left(-a\right)\right) \cdot \frac{a \cdot \left(\left(\left(-a\right) \cdot \left(b \cdot b\right)\right) \cdot 8\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right)}}\right)}} \]
      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 0 62.3

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

   4. Simplified 62.1

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

      [Start] 62.3	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot {x-scale}^{2}}{a \cdot \left(b \cdot \sqrt{8}\right)} \cdot \sqrt{\frac{1}{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right)} \]
      rational.json-simplify-43 [=>] 62.3	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot {x-scale}^{2}}{\color{blue}{b \cdot \left(\sqrt{8} \cdot a\right)}} \cdot \sqrt{\frac{1}{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right)} \]
      rational.json-simplify-46 [=>] 62.3	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\color{blue}{\frac{\frac{y-scale \cdot {x-scale}^{2}}{b}}{\sqrt{8} \cdot a}} \cdot \sqrt{\frac{1}{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right)} \]
      exponential.json-simplify-27 [=>] 62.3	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{\frac{y-scale \cdot {x-scale}^{2}}{b}}{\sqrt{8} \cdot a} \cdot \sqrt{\frac{1}{2 \cdot \color{blue}{{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right)} \]
      exponential.json-simplify-27 [=>] 62.1	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{\frac{y-scale \cdot {x-scale}^{2}}{b}}{\sqrt{8} \cdot a} \cdot \sqrt{\frac{1}{2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + 2 \cdot \color{blue}{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}}}\right)\right)} \]

   5. Taylor expanded in b around -inf 51.8

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

   6. Simplified 51.7

      \[\leadsto \color{blue}{y-scale \cdot \left(\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\right) \cdot -0.25\right)} \]
      Proof

      [Start] 51.8	\[ -0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right) \]
      rational.json-simplify-43 [=>] 51.8	\[ \color{blue}{y-scale \cdot \left(\left(\sqrt{2} \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right) \cdot -0.25\right)} \]
      rational.json-simplify-43 [=>] 51.8	\[ y-scale \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sqrt{8} \cdot b\right)\right)}\right) \cdot -0.25\right) \]
      rational.json-simplify-2 [<=] 51.8	\[ y-scale \cdot \left(\left(\sqrt{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(b \cdot \sqrt{8}\right)}\right)\right) \cdot -0.25\right) \]
      rational.json-simplify-43 [=>] 51.8	\[ y-scale \cdot \left(\color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right)} \cdot -0.25\right) \]
      rational.json-simplify-43 [=>] 51.8	\[ y-scale \cdot \left(\left(\cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right) \cdot -0.25\right) \]
      rational.json-simplify-2 [<=] 51.8	\[ y-scale \cdot \left(\left(\cos \left(angle \cdot \color{blue}{\left(0.005555555555555556 \cdot \pi\right)}\right) \cdot \left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right) \cdot -0.25\right) \]
      rational.json-simplify-2 [<=] 51.8	\[ y-scale \cdot \left(\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)}\right) \cdot -0.25\right) \]
      rational.json-simplify-43 [=>] 51.9	\[ y-scale \cdot \left(\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \color{blue}{\left(b \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}\right) \cdot -0.25\right) \]
      exponential.json-simplify-20 [=>] 51.7	\[ y-scale \cdot \left(\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot \color{blue}{\sqrt{2 \cdot 8}}\right)\right) \cdot -0.25\right) \]
      metadata-eval [=>] 51.7	\[ y-scale \cdot \left(\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot \sqrt{\color{blue}{16}}\right)\right) \cdot -0.25\right) \]
      metadata-eval [=>] 51.7	\[ y-scale \cdot \left(\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot \color{blue}{4}\right)\right) \cdot -0.25\right) \]

   if -2.45000000000000005e-144 < x-scale < -1.7e-252

   1. Initial program 63.9

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

   2. Simplified 63.9

      \[\leadsto \color{blue}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \frac{b \cdot a}{\sqrt{\left(b \cdot a\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2} + {\left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \frac{\frac{2}{x-scale}}{y-scale}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)}^{2}}\right)\right) \cdot \left(\left(b \cdot \left(-a\right)\right) \cdot \frac{a \cdot \left(\left(\left(-a\right) \cdot \left(b \cdot b\right)\right) \cdot 8\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right)}}\right)}} \]
      Proof

      [Start] 63.9

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

   3. Taylor expanded in x-scale around 0 63.6

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

   4. Simplified 63.6

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

      [Start] 63.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot {x-scale}^{2}}{a \cdot \left(b \cdot \sqrt{8}\right)} \cdot \sqrt{\frac{1}{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right)} \]
      rational.json-simplify-43 [=>] 63.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot {x-scale}^{2}}{\color{blue}{b \cdot \left(\sqrt{8} \cdot a\right)}} \cdot \sqrt{\frac{1}{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right)} \]
      rational.json-simplify-46 [=>] 63.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\color{blue}{\frac{\frac{y-scale \cdot {x-scale}^{2}}{b}}{\sqrt{8} \cdot a}} \cdot \sqrt{\frac{1}{2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right)} \]
      exponential.json-simplify-27 [=>] 63.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{\frac{y-scale \cdot {x-scale}^{2}}{b}}{\sqrt{8} \cdot a} \cdot \sqrt{\frac{1}{2 \cdot \color{blue}{{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right)} \]
      exponential.json-simplify-27 [=>] 63.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{\frac{y-scale \cdot {x-scale}^{2}}{b}}{\sqrt{8} \cdot a} \cdot \sqrt{\frac{1}{2 \cdot {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + 2 \cdot \color{blue}{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}}}\right)\right)} \]

   5. Taylor expanded in b around inf 49.6

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

   6. Simplified 49.4

      \[\leadsto \color{blue}{y-scale \cdot \left(0.25 \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\right)\right)} \]
      Proof

      [Start] 49.6	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right) \]
      rational.json-simplify-43 [=>] 49.6	\[ \color{blue}{y-scale \cdot \left(\left(\sqrt{2} \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right) \cdot 0.25\right)} \]
      rational.json-simplify-2 [=>] 49.6	\[ y-scale \cdot \color{blue}{\left(0.25 \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)} \]
      rational.json-simplify-43 [=>] 49.6	\[ y-scale \cdot \left(0.25 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sqrt{8} \cdot b\right)\right)}\right)\right) \]
      rational.json-simplify-2 [<=] 49.6	\[ y-scale \cdot \left(0.25 \cdot \left(\sqrt{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(b \cdot \sqrt{8}\right)}\right)\right)\right) \]
      rational.json-simplify-43 [=>] 49.6	\[ y-scale \cdot \left(0.25 \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right)}\right) \]
      rational.json-simplify-43 [=>] 49.5	\[ y-scale \cdot \left(0.25 \cdot \left(\cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)} \cdot \left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right)\right) \]
      rational.json-simplify-2 [<=] 49.5	\[ y-scale \cdot \left(0.25 \cdot \left(\cos \left(angle \cdot \color{blue}{\left(0.005555555555555556 \cdot \pi\right)}\right) \cdot \left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right)\right) \]
      rational.json-simplify-2 [<=] 49.5	\[ y-scale \cdot \left(0.25 \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)}\right)\right) \]
      rational.json-simplify-43 [=>] 49.6	\[ y-scale \cdot \left(0.25 \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \color{blue}{\left(b \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)}\right)\right) \]
      exponential.json-simplify-20 [=>] 49.4	\[ y-scale \cdot \left(0.25 \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot \color{blue}{\sqrt{2 \cdot 8}}\right)\right)\right) \]
      metadata-eval [=>] 49.4	\[ y-scale \cdot \left(0.25 \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot \sqrt{\color{blue}{16}}\right)\right)\right) \]
      metadata-eval [=>] 49.4	\[ y-scale \cdot \left(0.25 \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot \color{blue}{4}\right)\right)\right) \]

   if 1.15000000000000007e-198 < x-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 63.1

      \[\leadsto \color{blue}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \frac{b \cdot a}{\sqrt{\left(b \cdot a\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2} + {\left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(\left({b}^{2} - {a}^{2}\right) \cdot \frac{\frac{2}{x-scale}}{y-scale}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\right)}^{2}}\right)\right) \cdot \left(\left(b \cdot \left(-a\right)\right) \cdot \frac{a \cdot \left(\left(\left(-a\right) \cdot \left(b \cdot b\right)\right) \cdot 8\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right)}}\right)}} \]
      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 x-scale around inf 62.6

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

   4. Simplified 62.5

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

      [Start] 62.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot x-scale}{a \cdot \left(b \cdot \sqrt{8}\right)} \cdot \sqrt{\frac{1}{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right)\right)} \]
      rational.json-simplify-43 [=>] 62.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot x-scale}{\color{blue}{b \cdot \left(\sqrt{8} \cdot a\right)}} \cdot \sqrt{\frac{1}{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right)\right)} \]
      exponential.json-simplify-27 [=>] 62.6	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot x-scale}{b \cdot \left(\sqrt{8} \cdot a\right)} \cdot \sqrt{\frac{1}{2 \cdot \frac{\color{blue}{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right)\right)} \]
      exponential.json-simplify-27 [=>] 62.5	\[ \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{4 \cdot \left(\left(b \cdot a\right) \cdot \left(\frac{y-scale \cdot x-scale}{b \cdot \left(\sqrt{8} \cdot a\right)} \cdot \sqrt{\frac{1}{2 \cdot \frac{{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{\color{blue}{{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}}{{y-scale}^{2}}}}\right)\right)} \]

   5. Taylor expanded in y-scale around 0 48.9

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

   6. Simplified 46.7

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

      [Start] 48.9	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      rational.json-simplify-2 [=>] 48.9	\[ 0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)} \]
      rational.json-simplify-43 [=>] 48.8	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
      exponential.json-simplify-20 [=>] 48.8	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\sqrt{\left(2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right) \cdot 8}}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 47.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -9.5 \cdot 10^{+54}:\\ \;\;\;\;\left(x-scale \cdot \sqrt{\left(2 \cdot \left({\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\right)\right) \cdot 8}\right) \cdot -0.25\\ \mathbf{elif}\;x-scale \leq -2.45 \cdot 10^{-144}:\\ \;\;\;\;y-scale \cdot \left(\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq -1.7 \cdot 10^{-252}:\\ \;\;\;\;y-scale \cdot \left(0.25 \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.15 \cdot 10^{-198}:\\ \;\;\;\;y-scale \cdot \left(\left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \sqrt{\left(2 \cdot \left({\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\right)\right) \cdot 8}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	49.4
Cost	46800

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\\ t_2 := y-scale \cdot \left(t_1 \cdot -0.25\right)\\ \mathbf{if}\;x-scale \leq -9.5 \cdot 10^{+54}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \mathbf{elif}\;x-scale \leq -8.4 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq -6.5 \cdot 10^{-251}:\\ \;\;\;\;y-scale \cdot \left(0.25 \cdot t_1\right)\\ \mathbf{elif}\;x-scale \leq 6 \cdot 10^{-198}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \sqrt{\left(2 \cdot \left({\left(a \cdot \cos t_0\right)}^{2} + {\left(b \cdot \sin t_0\right)}^{2}\right)\right) \cdot 8}\right)\\ \end{array} \]

Alternative 2
Error	49.4
Cost	40336

\[\begin{array}{l} t_0 := \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\\ t_1 := y-scale \cdot \left(t_0 \cdot -0.25\right)\\ \mathbf{if}\;x-scale \leq -1.05 \cdot 10^{+55}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \mathbf{elif}\;x-scale \leq -7.4 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -1.18 \cdot 10^{-250}:\\ \;\;\;\;y-scale \cdot \left(0.25 \cdot t_0\right)\\ \mathbf{elif}\;x-scale \leq 4.2 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot {\left(a \cdot 1\right)}^{2} + 2 \cdot {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right)\\ \end{array} \]

Alternative 3
Error	51.4
Cost	14096

\[\begin{array}{l} t_0 := \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\\ t_1 := x-scale \cdot \left(-a\right)\\ t_2 := y-scale \cdot \left(t_0 \cdot -0.25\right)\\ \mathbf{if}\;x-scale \leq -1.05 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -1.85 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq -1.28 \cdot 10^{-250}:\\ \;\;\;\;y-scale \cdot \left(0.25 \cdot t_0\right)\\ \mathbf{elif}\;x-scale \leq 3.4 \cdot 10^{-19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	51.6
Cost	1100

\[\begin{array}{l} t_0 := x-scale \cdot \left(-a\right)\\ \mathbf{if}\;x-scale \leq -5.3 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -7.6 \cdot 10^{-137}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;x-scale \leq 4.7 \cdot 10^{-203}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 3\right) + y-scale \cdot b\right)\\ \mathbf{elif}\;x-scale \leq 4.5 \cdot 10^{-28}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	51.6
Cost	844

\[\begin{array}{l} t_0 := x-scale \cdot \left(-a\right)\\ \mathbf{if}\;x-scale \leq -1.06 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -1.05 \cdot 10^{-137}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;x-scale \leq 8 \cdot 10^{-204}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.2 \cdot 10^{-31}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	53.9
Cost	520

\[\begin{array}{l} t_0 := x-scale \cdot \left(-a\right)\\ \mathbf{if}\;x-scale \leq -5.2 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 2.3 \cdot 10^{-23}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	54.2
Cost	192

\[x-scale \cdot a \]

Reproduce

herbie shell --seed 2023067 
(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))))