a from scale-rotated-ellipse

Average Error: 63.4 → 47.0

Time: 1.3min

Precision: binary64

Cost: 72260

Math

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

↓

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;x-scale \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos t_0}^{2} + {b}^{2} \cdot {\sin t_0}^{2}}\right)\right)\\ \mathbf{elif}\;x-scale \leq 8.2 \cdot 10^{-104}:\\ \;\;\;\;0.25 \cdot \left(-\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b \cdot \left(4 \cdot y-scale\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(\cos t_1 \cdot a\right)}^{2} + {\left(\sin t_1 \cdot b\right)}^{2}\right)\right)}\right)\\ \end{array} \]

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 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* PI (* 0.005555555555555556 angle))))
   (if (<= x-scale -3.3e-93)
     (*
      0.25
      (*
       -1.0
       (*
        (* (sqrt 2.0) (* x-scale (sqrt 8.0)))
        (sqrt
         (+
          (* (pow a 2.0) (pow (cos t_0) 2.0))
          (* (pow b 2.0) (pow (sin t_0) 2.0)))))))
     (if (<= x-scale 8.2e-104)
       (*
        0.25
        (-
         (*
          (cos (* PI (* angle 0.005555555555555556)))
          (* b (* 4.0 y-scale)))))
       (*
        0.25
        (*
         x-scale
         (sqrt
          (*
           8.0
           (*
            2.0
            (+ (pow (* (cos t_1) a) 2.0) (pow (* (sin t_1) b) 2.0)))))))))))

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 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (x_45_scale <= -3.3e-93) {
		tmp = 0.25 * (-1.0 * ((sqrt(2.0) * (x_45_scale * sqrt(8.0))) * sqrt(((pow(a, 2.0) * pow(cos(t_0), 2.0)) + (pow(b, 2.0) * pow(sin(t_0), 2.0))))));
	} else if (x_45_scale <= 8.2e-104) {
		tmp = 0.25 * -(cos((((double) M_PI) * (angle * 0.005555555555555556))) * (b * (4.0 * y_45_scale)));
	} else {
		tmp = 0.25 * (x_45_scale * sqrt((8.0 * (2.0 * (pow((cos(t_1) * a), 2.0) + pow((sin(t_1) * b), 2.0))))));
	}
	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 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (x_45_scale <= -3.3e-93) {
		tmp = 0.25 * (-1.0 * ((Math.sqrt(2.0) * (x_45_scale * Math.sqrt(8.0))) * Math.sqrt(((Math.pow(a, 2.0) * Math.pow(Math.cos(t_0), 2.0)) + (Math.pow(b, 2.0) * Math.pow(Math.sin(t_0), 2.0))))));
	} else if (x_45_scale <= 8.2e-104) {
		tmp = 0.25 * -(Math.cos((Math.PI * (angle * 0.005555555555555556))) * (b * (4.0 * y_45_scale)));
	} else {
		tmp = 0.25 * (x_45_scale * Math.sqrt((8.0 * (2.0 * (Math.pow((Math.cos(t_1) * a), 2.0) + Math.pow((Math.sin(t_1) * b), 2.0))))));
	}
	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 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.pi * (0.005555555555555556 * angle)
	tmp = 0
	if x_45_scale <= -3.3e-93:
		tmp = 0.25 * (-1.0 * ((math.sqrt(2.0) * (x_45_scale * math.sqrt(8.0))) * math.sqrt(((math.pow(a, 2.0) * math.pow(math.cos(t_0), 2.0)) + (math.pow(b, 2.0) * math.pow(math.sin(t_0), 2.0))))))
	elif x_45_scale <= 8.2e-104:
		tmp = 0.25 * -(math.cos((math.pi * (angle * 0.005555555555555556))) * (b * (4.0 * y_45_scale)))
	else:
		tmp = 0.25 * (x_45_scale * math.sqrt((8.0 * (2.0 * (math.pow((math.cos(t_1) * a), 2.0) + math.pow((math.sin(t_1) * b), 2.0))))))
	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(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (x_45_scale <= -3.3e-93)
		tmp = Float64(0.25 * Float64(-1.0 * Float64(Float64(sqrt(2.0) * Float64(x_45_scale * sqrt(8.0))) * sqrt(Float64(Float64((a ^ 2.0) * (cos(t_0) ^ 2.0)) + Float64((b ^ 2.0) * (sin(t_0) ^ 2.0)))))));
	elseif (x_45_scale <= 8.2e-104)
		tmp = Float64(0.25 * Float64(-Float64(cos(Float64(pi * Float64(angle * 0.005555555555555556))) * Float64(b * Float64(4.0 * y_45_scale)))));
	else
		tmp = Float64(0.25 * Float64(x_45_scale * sqrt(Float64(8.0 * Float64(2.0 * Float64((Float64(cos(t_1) * a) ^ 2.0) + (Float64(sin(t_1) * b) ^ 2.0)))))));
	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 = 0.005555555555555556 * (angle * pi);
	t_1 = pi * (0.005555555555555556 * angle);
	tmp = 0.0;
	if (x_45_scale <= -3.3e-93)
		tmp = 0.25 * (-1.0 * ((sqrt(2.0) * (x_45_scale * sqrt(8.0))) * sqrt((((a ^ 2.0) * (cos(t_0) ^ 2.0)) + ((b ^ 2.0) * (sin(t_0) ^ 2.0))))));
	elseif (x_45_scale <= 8.2e-104)
		tmp = 0.25 * -(cos((pi * (angle * 0.005555555555555556))) * (b * (4.0 * y_45_scale)));
	else
		tmp = 0.25 * (x_45_scale * sqrt((8.0 * (2.0 * (((cos(t_1) * a) ^ 2.0) + ((sin(t_1) * b) ^ 2.0))))));
	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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -3.3e-93], N[(0.25 * N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 8.2e-104], N[(0.25 * (-N[(N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b * N[(4.0 * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(0.25 * N[(x$45$scale * N[Sqrt[N[(8.0 * N[(2.0 * N[(N[Power[N[(N[Cos[t$95$1], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[t$95$1], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -3.3000000000000001e-93

   1. Initial program 63.2

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

   2. Simplified 63.1

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

      [Start] 63.2

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

   3. Taylor expanded in x-scale around inf 60.2

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

   4. Simplified 59.8

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

      [Start] 60.2	\[ 0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      rational_best-simplify-47 [=>] 60.2	\[ 0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}}\right) \]
      rational_best-simplify-1 [=>] 60.2	\[ 0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}}\right) \]
      exponential-simplify-27 [=>] 60.2	\[ 0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)}^{2}}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right) \]

   5. Taylor expanded in y-scale around -inf 46.3

      \[\leadsto 0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)} \]

   if -3.3000000000000001e-93 < x-scale < 8.19999999999999968e-104

   1. Initial program 63.8

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

   2. Simplified 63.8

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

      [Start] 63.8

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

   3. Taylor expanded in x-scale around 0 53.1

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

   4. Simplified 51.3

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

      [Start] 53.1	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right) \]
      rational_best-simplify-2 [<=] 53.1	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right) \]
      exponential-simplify-27 [=>] 53.1	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \left(2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right) \]
      rational_best-simplify-1 [=>] 53.1	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + \color{blue}{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}}\right) \]
      exponential-simplify-27 [=>] 53.1	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + \left(\color{blue}{{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right) \]

   5. Taylor expanded in angle around inf 53.1

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

   6. Simplified 51.3

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

      [Start] 53.1	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right) \]
      rational_best-simplify-1 [=>] 53.1	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \]
      rational_best-simplify-2 [=>] 53.1	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} + 2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      rational_best-simplify-47 [=>] 53.1	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \]
      exponential-simplify-27 [=>] 51.3	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left(\color{blue}{{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)}^{2}} + {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      trig-simplify-13 [=>] 51.3	\[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)}^{2}\right)}}\right) \]

   7. Taylor expanded in b around -inf 50.0

      \[\leadsto 0.25 \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)\right)} \]

   8. Simplified 49.8

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

      [Start] 50.0	\[ 0.25 \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)\right) \]
      rational_best-simplify-2 [=>] 50.0	\[ 0.25 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right) \cdot -1\right)} \]
      rational_best-simplify-44 [<=] 50.0	\[ 0.25 \cdot \left(\color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)} \cdot -1\right) \]
      rational_best-simplify-12 [=>] 50.0	\[ 0.25 \cdot \color{blue}{\left(-y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)} \]
      rational_best-simplify-2 [=>] 50.0	\[ 0.25 \cdot \left(-y-scale \cdot \color{blue}{\left(\left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}\right)}\right) \]
      rational_best-simplify-44 [<=] 50.0	\[ 0.25 \cdot \left(-y-scale \cdot \left(\color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot \sqrt{8}\right)\right)} \cdot \sqrt{2}\right)\right) \]
      rational_best-simplify-2 [<=] 50.0	\[ 0.25 \cdot \left(-y-scale \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot \sqrt{8}\right)\right)\right)}\right) \]
      rational_best-simplify-44 [=>] 50.0	\[ 0.25 \cdot \left(-y-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right)}\right) \]
      rational_best-simplify-44 [=>] 50.0	\[ 0.25 \cdot \left(-\color{blue}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right)}\right) \]

   if 8.19999999999999968e-104 < x-scale

   1. Initial program 63.1

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

   2. Simplified 63.0

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

      [Start] 63.1

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

   3. Taylor expanded in y-scale around 0 46.9

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

   4. Simplified 44.1

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

      [Start] 46.9	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      rational_best-simplify-2 [=>] 46.9	\[ 0.25 \cdot \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)} \]
      rational_best-simplify-44 [=>] 46.9	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)} \cdot \sqrt{8}\right)\right)} \]
      exponential-simplify-19 [=>] 46.8	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\sqrt{8 \cdot \left(2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}}\right) \]
      rational_best-simplify-47 [=>] 46.8	\[ 0.25 \cdot \left(x-scale \cdot \sqrt{8 \cdot \color{blue}{\left(2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 47.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \mathbf{elif}\;x-scale \leq 8.2 \cdot 10^{-104}:\\ \;\;\;\;0.25 \cdot \left(-\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b \cdot \left(4 \cdot y-scale\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left(2 \cdot \left({\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}^{2} + {\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	51.4
Cost	14292

\[\begin{array}{l} t_0 := 0.25 \cdot \left(-x-scale \cdot \left(4 \cdot a\right)\right)\\ \mathbf{if}\;b \leq -105000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -8.6 \cdot 10^{-283}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(4 \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.76 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+137}:\\ \;\;\;\;0.25 \cdot \left(-b \cdot \left(y-scale \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 2
Error	51.4
Cost	14292

\[\begin{array}{l} t_0 := \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot 4\\ \mathbf{if}\;b \leq -92000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-224}:\\ \;\;\;\;0.25 \cdot \left(-x-scale \cdot \left(4 \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-279}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(4 \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-180}:\\ \;\;\;\;0.25 \cdot \left(-x-scale \cdot \left(a \cdot t_0\right)\right)\\ \mathbf{elif}\;b \leq 4.25 \cdot 10^{+139}:\\ \;\;\;\;0.25 \cdot \left(-b \cdot \left(y-scale \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 3
Error	51.5
Cost	14292

\[\begin{array}{l} \mathbf{if}\;b \leq -1250000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq -8.8 \cdot 10^{-225}:\\ \;\;\;\;0.25 \cdot \left(-x-scale \cdot \left(4 \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-279}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(4 \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-181}:\\ \;\;\;\;0.25 \cdot \left(-x-scale \cdot \left(a \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot 4\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+135}:\\ \;\;\;\;0.25 \cdot \left(-\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(b \cdot \left(4 \cdot y-scale\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 4
Error	51.4
Cost	14228

\[\begin{array}{l} t_0 := 0.25 \cdot \left(-x-scale \cdot \left(4 \cdot a\right)\right)\\ \mathbf{if}\;b \leq -250000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-282}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(4 \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+135}:\\ \;\;\;\;0.25 \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(y-scale \cdot \left(b \cdot -4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 5
Error	51.4
Cost	13964

\[\begin{array}{l} t_0 := 0.25 \cdot \left(-x-scale \cdot \left(4 \cdot a\right)\right)\\ \mathbf{if}\;b \leq -220000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq -1.95 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.06 \cdot 10^{-277}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(a \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot 4\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 6
Error	51.4
Cost	13964

\[\begin{array}{l} t_0 := 0.25 \cdot \left(-x-scale \cdot \left(4 \cdot a\right)\right)\\ \mathbf{if}\;b \leq -1300000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-228}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-282}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(4 \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 10^{+139}:\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 7
Error	51.4
Cost	1040

\[\begin{array}{l} t_0 := 0.25 \cdot \left(-x-scale \cdot \left(4 \cdot a\right)\right)\\ \mathbf{if}\;b \leq -175000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-280}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(a \cdot 4\right)\right)\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+136}:\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 8
Error	51.6
Cost	844

\[\begin{array}{l} t_0 := y-scale \cdot \left(-b\right)\\ \mathbf{if}\;b \leq -0.72:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq -7.5 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-126}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(a \cdot 4\right)\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 9
Error	53.8
Cost	520

\[\begin{array}{l} \mathbf{if}\;b \leq -260:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+136}:\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 10
Error	53.9
Cost	192

\[y-scale \cdot b \]

Reproduce

herbie shell --seed 2023092 
(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))))