a from scale-rotated-ellipse

Average Error: 99.1% → 71.65%

Time: 1.9min

Precision: binary64

Cost: 27156

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\\ t_1 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ t_2 := -0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \left(\sqrt{8} \cdot t_0\right)\right)\right)\right)\\ \mathbf{if}\;x-scale \leq -4.4 \cdot 10^{+117}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \left(a \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq -2.8 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -1.18 \cdot 10^{-198}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;x-scale \leq -5.8 \cdot 10^{-202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x-scale \leq 1.8 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right)\right)\right)\\ \end{array} \]

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 (cos (* 0.005555555555555556 (* PI angle))))
        (t_1 (* (* y-scale 0.25) (* b -4.0)))
        (t_2 (* -0.25 (* (sqrt 2.0) (* x-scale (* a (* (sqrt 8.0) t_0)))))))
   (if (<= x-scale -4.4e+117)
     (* 0.25 (* x-scale (* 4.0 (* a t_0))))
     (if (<= x-scale -1.2e-5)
       t_2
       (if (<= x-scale -2.8e-27)
         t_1
         (if (<= x-scale -1.18e-198)
           (* (* y-scale 0.25) (* b 4.0))
           (if (<= x-scale -5.8e-202)
             t_2
             (if (<= x-scale 1.8e-181)
               t_1
               (*
                0.25
                (*
                 x-scale
                 (*
                  4.0
                  (hypot
                   (* b (sin (* (* 0.005555555555555556 PI) angle)))
                   a))))))))))))

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 = cos((0.005555555555555556 * (((double) M_PI) * angle)));
	double t_1 = (y_45_scale * 0.25) * (b * -4.0);
	double t_2 = -0.25 * (sqrt(2.0) * (x_45_scale * (a * (sqrt(8.0) * t_0))));
	double tmp;
	if (x_45_scale <= -4.4e+117) {
		tmp = 0.25 * (x_45_scale * (4.0 * (a * t_0)));
	} else if (x_45_scale <= -1.2e-5) {
		tmp = t_2;
	} else if (x_45_scale <= -2.8e-27) {
		tmp = t_1;
	} else if (x_45_scale <= -1.18e-198) {
		tmp = (y_45_scale * 0.25) * (b * 4.0);
	} else if (x_45_scale <= -5.8e-202) {
		tmp = t_2;
	} else if (x_45_scale <= 1.8e-181) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (x_45_scale * (4.0 * hypot((b * sin(((0.005555555555555556 * ((double) M_PI)) * angle))), a)));
	}
	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.cos((0.005555555555555556 * (Math.PI * angle)));
	double t_1 = (y_45_scale * 0.25) * (b * -4.0);
	double t_2 = -0.25 * (Math.sqrt(2.0) * (x_45_scale * (a * (Math.sqrt(8.0) * t_0))));
	double tmp;
	if (x_45_scale <= -4.4e+117) {
		tmp = 0.25 * (x_45_scale * (4.0 * (a * t_0)));
	} else if (x_45_scale <= -1.2e-5) {
		tmp = t_2;
	} else if (x_45_scale <= -2.8e-27) {
		tmp = t_1;
	} else if (x_45_scale <= -1.18e-198) {
		tmp = (y_45_scale * 0.25) * (b * 4.0);
	} else if (x_45_scale <= -5.8e-202) {
		tmp = t_2;
	} else if (x_45_scale <= 1.8e-181) {
		tmp = t_1;
	} else {
		tmp = 0.25 * (x_45_scale * (4.0 * Math.hypot((b * Math.sin(((0.005555555555555556 * Math.PI) * angle))), a)));
	}
	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.cos((0.005555555555555556 * (math.pi * angle)))
	t_1 = (y_45_scale * 0.25) * (b * -4.0)
	t_2 = -0.25 * (math.sqrt(2.0) * (x_45_scale * (a * (math.sqrt(8.0) * t_0))))
	tmp = 0
	if x_45_scale <= -4.4e+117:
		tmp = 0.25 * (x_45_scale * (4.0 * (a * t_0)))
	elif x_45_scale <= -1.2e-5:
		tmp = t_2
	elif x_45_scale <= -2.8e-27:
		tmp = t_1
	elif x_45_scale <= -1.18e-198:
		tmp = (y_45_scale * 0.25) * (b * 4.0)
	elif x_45_scale <= -5.8e-202:
		tmp = t_2
	elif x_45_scale <= 1.8e-181:
		tmp = t_1
	else:
		tmp = 0.25 * (x_45_scale * (4.0 * math.hypot((b * math.sin(((0.005555555555555556 * math.pi) * angle))), a)))
	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 = cos(Float64(0.005555555555555556 * Float64(pi * angle)))
	t_1 = Float64(Float64(y_45_scale * 0.25) * Float64(b * -4.0))
	t_2 = Float64(-0.25 * Float64(sqrt(2.0) * Float64(x_45_scale * Float64(a * Float64(sqrt(8.0) * t_0)))))
	tmp = 0.0
	if (x_45_scale <= -4.4e+117)
		tmp = Float64(0.25 * Float64(x_45_scale * Float64(4.0 * Float64(a * t_0))));
	elseif (x_45_scale <= -1.2e-5)
		tmp = t_2;
	elseif (x_45_scale <= -2.8e-27)
		tmp = t_1;
	elseif (x_45_scale <= -1.18e-198)
		tmp = Float64(Float64(y_45_scale * 0.25) * Float64(b * 4.0));
	elseif (x_45_scale <= -5.8e-202)
		tmp = t_2;
	elseif (x_45_scale <= 1.8e-181)
		tmp = t_1;
	else
		tmp = Float64(0.25 * Float64(x_45_scale * Float64(4.0 * hypot(Float64(b * sin(Float64(Float64(0.005555555555555556 * pi) * angle))), a))));
	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 = cos((0.005555555555555556 * (pi * angle)));
	t_1 = (y_45_scale * 0.25) * (b * -4.0);
	t_2 = -0.25 * (sqrt(2.0) * (x_45_scale * (a * (sqrt(8.0) * t_0))));
	tmp = 0.0;
	if (x_45_scale <= -4.4e+117)
		tmp = 0.25 * (x_45_scale * (4.0 * (a * t_0)));
	elseif (x_45_scale <= -1.2e-5)
		tmp = t_2;
	elseif (x_45_scale <= -2.8e-27)
		tmp = t_1;
	elseif (x_45_scale <= -1.18e-198)
		tmp = (y_45_scale * 0.25) * (b * 4.0);
	elseif (x_45_scale <= -5.8e-202)
		tmp = t_2;
	elseif (x_45_scale <= 1.8e-181)
		tmp = t_1;
	else
		tmp = 0.25 * (x_45_scale * (4.0 * hypot((b * sin(((0.005555555555555556 * pi) * angle))), a)));
	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[Cos[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$45$scale * 0.25), $MachinePrecision] * N[(b * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x$45$scale * N[(a * N[(N[Sqrt[8.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -4.4e+117], N[(0.25 * N[(x$45$scale * N[(4.0 * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -1.2e-5], t$95$2, If[LessEqual[x$45$scale, -2.8e-27], t$95$1, If[LessEqual[x$45$scale, -1.18e-198], N[(N[(y$45$scale * 0.25), $MachinePrecision] * N[(b * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -5.8e-202], t$95$2, If[LessEqual[x$45$scale, 1.8e-181], t$95$1, N[(0.25 * N[(x$45$scale * N[(4.0 * N[Sqrt[N[(b * N[Sin[N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + a ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x-scale < -4.40000000000000028e117

   1. Initial program 99.46

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

   2. Simplified 99

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

      [Start] 99.46

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

   3. Taylor expanded in y-scale around 0 99.01

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

   4. Simplified 99

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

      [Start] 99.01	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      associate-*l* [=>] 99	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
      distribute-lft-out [=>] 99	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      fma-def [=>] 99	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 99	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{a \cdot a}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right) \]
      *-commutative [=>] 99	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \color{blue}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 99	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \color{blue}{\left(b \cdot b\right)}\right)}\right)\right) \]

   5. Taylor expanded in a around inf 79.5

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

   6. Applied egg-rr 97.59

      \[\leadsto 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 4\right)} - 1\right)}\right) \]

   7. Simplified 79.29

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

      [Start] 97.59	\[ 0.25 \cdot \left(x-scale \cdot \left(e^{\mathsf{log1p}\left(\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 4\right)} - 1\right)\right) \]
      expm1-def [=>] 85.57	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 4\right)\right)}\right) \]
      expm1-log1p [=>] 79.29	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 4\right)}\right) \]
      *-commutative [=>] 79.29	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(4 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right) \]

   if -4.40000000000000028e117 < x-scale < -1.2e-5 or -1.18000000000000006e-198 < x-scale < -5.79999999999999976e-202

   1. Initial program 98.09

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

   2. Simplified 98.03

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

      [Start] 98.09

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

   3. Taylor expanded in a around -inf 93.27

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}\right)} \]

   4. Simplified 92.58

      \[\leadsto \color{blue}{-0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \sqrt{\left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} \cdot \frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, {\left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}\right)}}\right)\right)} \]
      Proof

      [Start] 93.27

      \[ -0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}\right) \]

      associate-*l* [=>] 92.58

      \[ -0.25 \cdot \color{blue}{\left(y-scale \cdot \left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}} + {\left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}\right)\right)} \]

   5. Taylor expanded in x-scale around inf 90.29

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

   6. Simplified 90.31

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

      [Start] 90.29	\[ -0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \sqrt{\left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right) \]
      *-commutative [=>] 90.29	\[ -0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \sqrt{\left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \frac{{\cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}^{2}}{{y-scale}^{2}}}\right)\right) \]
      associate-*r* [<=] 90.31	\[ -0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \sqrt{\left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \frac{{\cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}^{2}}{{y-scale}^{2}}}\right)\right) \]
      *-commutative [<=] 90.31	\[ -0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \sqrt{\left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \frac{{\cos \left(angle \cdot \color{blue}{\left(0.005555555555555556 \cdot \pi\right)}\right)}^{2}}{{y-scale}^{2}}}\right)\right) \]
      unpow2 [=>] 90.31	\[ -0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \sqrt{\left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{\color{blue}{y-scale \cdot y-scale}}}\right)\right) \]
      *-commutative [=>] 90.31	\[ -0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \sqrt{\left(\frac{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \frac{{\cos \left(angle \cdot \color{blue}{\left(\pi \cdot 0.005555555555555556\right)}\right)}^{2}}{y-scale \cdot y-scale}}\right)\right) \]

   7. Applied egg-rr 86.7

      \[\leadsto -0.25 \cdot \left(y-scale \cdot \left(\left(\left(x-scale \cdot a\right) \cdot \sqrt{8}\right) \cdot \color{blue}{\mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{x-scale}, \frac{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{y-scale}\right), \frac{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}{y-scale}\right)}\right)\right) \]

   8. Taylor expanded in y-scale around 0 82.78

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

   if -1.2e-5 < x-scale < -2.8e-27 or -5.79999999999999976e-202 < x-scale < 1.8e-181

   1. Initial program 99.73

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

   2. Simplified 99.76

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \frac{\sqrt{\left(b \cdot \left(\left(a \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \frac{8 \cdot \left(b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}{x-scale} \cdot \frac{\cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale}\right)\right)}}{b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)}\right)} \]
      Proof

      [Start] 99.73

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

   3. Taylor expanded in angle around 0 76.37

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

   4. Simplified 76.36

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

      [Start] 76.37	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 76.36	\[ \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 76.36	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]
      *-commutative [=>] 76.36	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left(\sqrt{8} \cdot b\right)} \cdot \sqrt{2}\right) \]

   5. Applied egg-rr 83.97

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

   6. Simplified 83.97

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\sqrt{\left(16 \cdot b\right) \cdot b}} \]
      Proof

      [Start] 83.97	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)} \]
      associate-*r* [=>] 83.97	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\color{blue}{\left(2 \cdot 8\right) \cdot \left(b \cdot b\right)}} \]
      associate-*r* [=>] 83.97	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\color{blue}{\left(\left(2 \cdot 8\right) \cdot b\right) \cdot b}} \]
      metadata-eval [=>] 83.97	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\left(\color{blue}{16} \cdot b\right) \cdot b} \]

   7. Taylor expanded in b around -inf 76.94

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(-4 \cdot b\right)} \]

   8. Simplified 76.94

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

      [Start] 76.94	\[ \left(0.25 \cdot y-scale\right) \cdot \left(-4 \cdot b\right) \]
      *-commutative [=>] 76.94	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(b \cdot -4\right)} \]

   if -2.8e-27 < x-scale < -1.18000000000000006e-198

   1. Initial program 99.19

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

   2. Simplified 98.48

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \frac{\sqrt{\left(b \cdot \left(\left(a \cdot \left(b \cdot \left(-a\right)\right)\right) \cdot \frac{8 \cdot \left(b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)\right) \cdot \left(\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right) + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{2 \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(b \cdot b - a \cdot a\right)\right)}{x-scale} \cdot \frac{\cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale}\right)\right)}}{b \cdot \left(\left(-a\right) \cdot \left(b \cdot a\right)\right)}\right)} \]
      Proof

      [Start] 99.19

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

   3. Taylor expanded in angle around 0 80.33

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

   4. Simplified 80.33

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

      [Start] 80.33	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 80.33	\[ \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 80.33	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]
      *-commutative [=>] 80.33	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left(\sqrt{8} \cdot b\right)} \cdot \sqrt{2}\right) \]

   5. Applied egg-rr 86.94

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

   6. Simplified 86.94

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\sqrt{\left(16 \cdot b\right) \cdot b}} \]
      Proof

      [Start] 86.94	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)} \]
      associate-*r* [=>] 86.94	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\color{blue}{\left(2 \cdot 8\right) \cdot \left(b \cdot b\right)}} \]
      associate-*r* [=>] 86.94	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\color{blue}{\left(\left(2 \cdot 8\right) \cdot b\right) \cdot b}} \]
      metadata-eval [=>] 86.94	\[ \left(0.25 \cdot y-scale\right) \cdot \sqrt{\left(\color{blue}{16} \cdot b\right) \cdot b} \]

   7. Taylor expanded in b around 0 80.14

      \[\leadsto \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(4 \cdot b\right)} \]

   if 1.8e-181 < x-scale

   1. Initial program 98.87

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

   2. Simplified 98.43

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

      [Start] 98.87

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

   3. Taylor expanded in y-scale around 0 75.62

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

   4. Simplified 75.6

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

      [Start] 75.62	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      associate-*l* [=>] 75.6	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
      distribute-lft-out [=>] 75.6	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      fma-def [=>] 75.6	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 75.6	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{a \cdot a}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right) \]
      *-commutative [=>] 75.6	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \color{blue}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 75.6	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \color{blue}{\left(b \cdot b\right)}\right)}\right)\right) \]

   5. Taylor expanded in angle around inf 75.68

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

   6. Simplified 59.63

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

      [Start] 75.68	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\right) \]
      +-commutative [=>] 75.68	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right)\right)\right) \]
      unpow2 [=>] 75.68	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\left(b \cdot b\right)} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\right) \]
      unpow2 [=>] 75.68	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \sqrt{\left(b \cdot b\right) \cdot \color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} + {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\right) \]
      swap-sqr [<=] 72.48	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} + {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\right) \]
      unpow2 [=>] 72.48	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \sqrt{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \color{blue}{\left(a \cdot a\right)} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\right) \]
      unpow2 [=>] 72.48	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \sqrt{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \left(a \cdot a\right) \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)\right)\right) \]
      swap-sqr [<=] 72.48	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \sqrt{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \color{blue}{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)\right)\right) \]

   7. Taylor expanded in angle around 0 59.43

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

   8. Applied egg-rr 74.88

      \[\leadsto 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), a\right) \cdot 4\right)} - 1\right)}\right) \]

   9. Simplified 59.03

      \[\leadsto 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right)\right)}\right) \]
      Proof

      [Start] 74.88	\[ 0.25 \cdot \left(x-scale \cdot \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), a\right) \cdot 4\right)} - 1\right)\right) \]
      expm1-def [=>] 60.59	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), a\right) \cdot 4\right)\right)}\right) \]
      expm1-log1p [=>] 59.07	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(\mathsf{hypot}\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), a\right) \cdot 4\right)}\right) \]
      *-commutative [=>] 59.07	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), a\right)\right)}\right) \]
      associate-*r* [=>] 59.03	\[ 0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \color{blue}{\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)}, a\right)\right)\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 71.65

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -4.4 \cdot 10^{+117}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.2 \cdot 10^{-5}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -2.8 \cdot 10^{-27}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;x-scale \leq -1.18 \cdot 10^{-198}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;x-scale \leq -5.8 \cdot 10^{-202}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.8 \cdot 10^{-181}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	76.01%
Cost	72076

\[\begin{array}{l} t_0 := \sqrt{8} \cdot \left(x-scale \cdot a\right)\\ t_1 := \left(0.005555555555555556 \cdot \pi\right) \cdot angle\\ t_2 := \cos t_1\\ t_3 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_4 := \frac{\cos t_3}{y-scale}\\ \mathbf{if}\;y-scale \leq -1.85 \cdot 10^{-96}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq 1.8 \cdot 10^{-207}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin t_1, a\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 5.6 \cdot 10^{-146}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{\sqrt[3]{{t_2}^{4}}} \cdot \left|\sqrt[3]{t_2}\right|\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 5.5 \cdot 10^{-85}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot t_0\right)\\ \mathbf{elif}\;y-scale \leq 4800000000 \lor \neg \left(y-scale \leq 3 \cdot 10^{+121}\right):\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|b \cdot \sqrt{8}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(t_0 \cdot \mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\sin t_3}{x-scale}, t_4\right), t_4\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	76.02%
Cost	60185

\[\begin{array}{l} t_0 := \sqrt{8} \cdot \left(x-scale \cdot a\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := \frac{\cos t_1}{y-scale}\\ \mathbf{if}\;y-scale \leq -5.2 \cdot 10^{-96}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq 1.8 \cdot 10^{-207}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 6.6 \cdot 10^{-147}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.25 \cdot 10^{-83}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot t_0\right)\\ \mathbf{elif}\;y-scale \leq 5600000000 \lor \neg \left(y-scale \leq 3 \cdot 10^{+121}\right):\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|b \cdot \sqrt{8}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(t_0 \cdot \mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\sin t_1}{x-scale}, t_2\right), t_2\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	71.47%
Cost	26892

\[\begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;x-scale \leq -8.6 \cdot 10^{+116}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \left(a \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.05 \cdot 10^{-65}:\\ \;\;\;\;-0.25 \cdot \left(a \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.85 \cdot 10^{-250}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\sqrt{8} \cdot t_0\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 2.5 \cdot 10^{-182}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	71.65%
Cost	20824

\[\begin{array}{l} t_0 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{if}\;x-scale \leq -7.4 \cdot 10^{+114}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -0.000115:\\ \;\;\;\;-0.25 \cdot \left(a \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.25 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -7.6 \cdot 10^{-199}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \mathbf{elif}\;x-scale \leq -5.8 \cdot 10^{-202}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 9 \cdot 10^{-182}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(4 \cdot \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	76.67%
Cost	20569

\[\begin{array}{l} t_0 := 0.25 \cdot \left(x-scale \cdot \left(4 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ t_1 := -0.25 \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\\ \mathbf{if}\;y-scale \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{elif}\;y-scale \leq -1.42 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 2.9 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 7.2 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 7.2 \cdot 10^{+74} \lor \neg \left(y-scale \leq 3 \cdot 10^{+121}\right):\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|b \cdot \sqrt{8}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	80.1%
Cost	14300

\[\begin{array}{l} t_0 := 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\right)\right)\\ t_1 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ t_2 := -0.25 \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\\ \mathbf{if}\;y-scale \leq -1.9 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -2.3 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 1.2 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 1.4 \cdot 10^{-146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 450000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 4 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{b \cdot \left(b \cdot 16\right)}\\ \end{array} \]

Alternative 7
Error	79.51%
Cost	14300

\[\begin{array}{l} t_0 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ t_1 := 0.25 \cdot \left(x-scale \cdot \left(4 \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ t_2 := -0.25 \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\\ \mathbf{if}\;y-scale \leq -1.4 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq -5.2 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 10^{-298}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 1.25 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 3.4 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 4500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 1.25 \cdot 10^{+122}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{b \cdot \left(b \cdot 16\right)}\\ \end{array} \]

Alternative 8
Error	80.09%
Cost	13904

\[\begin{array}{l} t_0 := -0.25 \cdot \left(a \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ t_1 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{if}\;y-scale \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 2.1 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 2050000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 8.5 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{b \cdot \left(b \cdot 16\right)}\\ \end{array} \]

Alternative 9
Error	80.1%
Cost	13904

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ t_1 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{if}\;y-scale \leq -1.7 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 2.4 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 6500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 8.1 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{b \cdot \left(b \cdot 16\right)}\\ \end{array} \]

Alternative 10
Error	80.1%
Cost	13904

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\\ t_1 := \left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{if}\;y-scale \leq -4.5 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 3 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 2300000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 4 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \sqrt{b \cdot \left(b \cdot 16\right)}\\ \end{array} \]

Alternative 11
Error	84.84%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{-157} \lor \neg \left(a \leq 4.3 \cdot 10^{-36}\right):\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot 0.25\right) \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 12
Error	83.6%
Cost	448

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

Reproduce

herbie shell --seed 2023090 
(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))))