a from scale-rotated-ellipse

Average Error: 63.5 → 41.8

Time: 1.7min

Precision: binary64

Cost: 53136

Math

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

↓

\[\begin{array}{l} t_0 := x-scale \cdot \sqrt{8}\\ t_1 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_2 := \sqrt{2} \cdot \left(a \cdot \cos t_1\right)\\ \mathbf{if}\;x-scale \leq -1.75 \cdot 10^{+61}:\\ \;\;\;\;\left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \left(x-scale \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq -7.2 \cdot 10^{-25}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -3.65 \cdot 10^{-112}:\\ \;\;\;\;-0.25 \cdot \left(t_0 \cdot t_2\right)\\ \mathbf{elif}\;x-scale \leq 1.8 \cdot 10^{-101}:\\ \;\;\;\;\left|\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(0.25 \cdot y-scale\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \mathsf{hypot}\left(t_2, \sqrt{2} \cdot \left(b \cdot \sin t_1\right)\right)\right)\\ \end{array} \]

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 (* x-scale (sqrt 8.0)))
        (t_1 (* 0.005555555555555556 (* PI angle)))
        (t_2 (* (sqrt 2.0) (* a (cos t_1)))))
   (if (<= x-scale -1.75e+61)
     (*
      (*
       a
       (*
        (sqrt 2.0)
        (* (sqrt 8.0) (cos (* angle (* PI 0.005555555555555556))))))
      (* x-scale -0.25))
     (if (<= x-scale -7.2e-25)
       (*
        (* x-scale 0.25)
        (*
         (sqrt 2.0)
         (* a (* (sqrt 8.0) (cos (* PI (* 0.005555555555555556 angle)))))))
       (if (<= x-scale -3.65e-112)
         (* -0.25 (* t_0 t_2))
         (if (<= x-scale 1.8e-101)
           (fabs (* (sqrt 8.0) (* (sqrt 2.0) (* b (* 0.25 y-scale)))))
           (* 0.25 (* t_0 (hypot t_2 (* (sqrt 2.0) (* b (sin t_1))))))))))))

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 = x_45_scale * sqrt(8.0);
	double t_1 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_2 = sqrt(2.0) * (a * cos(t_1));
	double tmp;
	if (x_45_scale <= -1.75e+61) {
		tmp = (a * (sqrt(2.0) * (sqrt(8.0) * cos((angle * (((double) M_PI) * 0.005555555555555556)))))) * (x_45_scale * -0.25);
	} else if (x_45_scale <= -7.2e-25) {
		tmp = (x_45_scale * 0.25) * (sqrt(2.0) * (a * (sqrt(8.0) * cos((((double) M_PI) * (0.005555555555555556 * angle))))));
	} else if (x_45_scale <= -3.65e-112) {
		tmp = -0.25 * (t_0 * t_2);
	} else if (x_45_scale <= 1.8e-101) {
		tmp = fabs((sqrt(8.0) * (sqrt(2.0) * (b * (0.25 * y_45_scale)))));
	} else {
		tmp = 0.25 * (t_0 * hypot(t_2, (sqrt(2.0) * (b * sin(t_1)))));
	}
	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 = x_45_scale * Math.sqrt(8.0);
	double t_1 = 0.005555555555555556 * (Math.PI * angle);
	double t_2 = Math.sqrt(2.0) * (a * Math.cos(t_1));
	double tmp;
	if (x_45_scale <= -1.75e+61) {
		tmp = (a * (Math.sqrt(2.0) * (Math.sqrt(8.0) * Math.cos((angle * (Math.PI * 0.005555555555555556)))))) * (x_45_scale * -0.25);
	} else if (x_45_scale <= -7.2e-25) {
		tmp = (x_45_scale * 0.25) * (Math.sqrt(2.0) * (a * (Math.sqrt(8.0) * Math.cos((Math.PI * (0.005555555555555556 * angle))))));
	} else if (x_45_scale <= -3.65e-112) {
		tmp = -0.25 * (t_0 * t_2);
	} else if (x_45_scale <= 1.8e-101) {
		tmp = Math.abs((Math.sqrt(8.0) * (Math.sqrt(2.0) * (b * (0.25 * y_45_scale)))));
	} else {
		tmp = 0.25 * (t_0 * Math.hypot(t_2, (Math.sqrt(2.0) * (b * Math.sin(t_1)))));
	}
	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 = x_45_scale * math.sqrt(8.0)
	t_1 = 0.005555555555555556 * (math.pi * angle)
	t_2 = math.sqrt(2.0) * (a * math.cos(t_1))
	tmp = 0
	if x_45_scale <= -1.75e+61:
		tmp = (a * (math.sqrt(2.0) * (math.sqrt(8.0) * math.cos((angle * (math.pi * 0.005555555555555556)))))) * (x_45_scale * -0.25)
	elif x_45_scale <= -7.2e-25:
		tmp = (x_45_scale * 0.25) * (math.sqrt(2.0) * (a * (math.sqrt(8.0) * math.cos((math.pi * (0.005555555555555556 * angle))))))
	elif x_45_scale <= -3.65e-112:
		tmp = -0.25 * (t_0 * t_2)
	elif x_45_scale <= 1.8e-101:
		tmp = math.fabs((math.sqrt(8.0) * (math.sqrt(2.0) * (b * (0.25 * y_45_scale)))))
	else:
		tmp = 0.25 * (t_0 * math.hypot(t_2, (math.sqrt(2.0) * (b * math.sin(t_1)))))
	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(x_45_scale * sqrt(8.0))
	t_1 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_2 = Float64(sqrt(2.0) * Float64(a * cos(t_1)))
	tmp = 0.0
	if (x_45_scale <= -1.75e+61)
		tmp = Float64(Float64(a * Float64(sqrt(2.0) * Float64(sqrt(8.0) * cos(Float64(angle * Float64(pi * 0.005555555555555556)))))) * Float64(x_45_scale * -0.25));
	elseif (x_45_scale <= -7.2e-25)
		tmp = Float64(Float64(x_45_scale * 0.25) * Float64(sqrt(2.0) * Float64(a * Float64(sqrt(8.0) * cos(Float64(pi * Float64(0.005555555555555556 * angle)))))));
	elseif (x_45_scale <= -3.65e-112)
		tmp = Float64(-0.25 * Float64(t_0 * t_2));
	elseif (x_45_scale <= 1.8e-101)
		tmp = abs(Float64(sqrt(8.0) * Float64(sqrt(2.0) * Float64(b * Float64(0.25 * y_45_scale)))));
	else
		tmp = Float64(0.25 * Float64(t_0 * hypot(t_2, Float64(sqrt(2.0) * Float64(b * sin(t_1))))));
	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 = x_45_scale * sqrt(8.0);
	t_1 = 0.005555555555555556 * (pi * angle);
	t_2 = sqrt(2.0) * (a * cos(t_1));
	tmp = 0.0;
	if (x_45_scale <= -1.75e+61)
		tmp = (a * (sqrt(2.0) * (sqrt(8.0) * cos((angle * (pi * 0.005555555555555556)))))) * (x_45_scale * -0.25);
	elseif (x_45_scale <= -7.2e-25)
		tmp = (x_45_scale * 0.25) * (sqrt(2.0) * (a * (sqrt(8.0) * cos((pi * (0.005555555555555556 * angle))))));
	elseif (x_45_scale <= -3.65e-112)
		tmp = -0.25 * (t_0 * t_2);
	elseif (x_45_scale <= 1.8e-101)
		tmp = abs((sqrt(8.0) * (sqrt(2.0) * (b * (0.25 * y_45_scale)))));
	else
		tmp = 0.25 * (t_0 * hypot(t_2, (sqrt(2.0) * (b * sin(t_1)))));
	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[(x$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(a * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.75e+61], N[(N[(a * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -7.2e-25], N[(N[(x$45$scale * 0.25), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a * N[(N[Sqrt[8.0], $MachinePrecision] * N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -3.65e-112], N[(-0.25 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 1.8e-101], N[Abs[N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(b * N[(0.25 * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.25 * N[(t$95$0 * N[Sqrt[t$95$2 ^ 2 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(b * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x-scale < -1.75000000000000009e61

   1. Initial program 63.3

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

   2. Simplified 63.1

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

      [Start] 63.3

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

   3. Taylor expanded in x-scale around inf 61.1

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

   4. Simplified 61.1

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

      [Start] 61.1	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(-1 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right)\right) \]
      mul-1-neg [=>] 61.1	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(-\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)}\right) \]
      *-commutative [=>] 61.1	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(-\color{blue}{\sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \cdot \frac{\sqrt{8}}{x-scale \cdot y-scale}}\right)\right) \]
      *-commutative [=>] 61.1	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(-\sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \cdot \frac{\sqrt{8}}{\color{blue}{y-scale \cdot x-scale}}\right)\right) \]

   5. Taylor expanded in a around -inf 51.2

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

   6. Simplified 51.2

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

      [Start] 51.2	\[ -0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right) \]
      associate-*r* [=>] 51.1	\[ \color{blue}{\left(-0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)} \]
      *-commutative [=>] 51.1	\[ \color{blue}{\left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right) \cdot \left(-0.25 \cdot x-scale\right)} \]
      associate-*r* [=>] 51.1	\[ \color{blue}{\left(\left(\sqrt{2} \cdot a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)} \cdot \left(-0.25 \cdot x-scale\right) \]
      *-commutative [=>] 51.1	\[ \left(\color{blue}{\left(a \cdot \sqrt{2}\right)} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right) \cdot \left(-0.25 \cdot x-scale\right) \]
      associate-*l* [=>] 51.2	\[ \color{blue}{\left(a \cdot \left(\sqrt{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)} \cdot \left(-0.25 \cdot x-scale\right) \]
      *-commutative [=>] 51.2	\[ \left(a \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)\right) \cdot \left(-0.25 \cdot x-scale\right) \]
      associate-*r* [=>] 51.2	\[ \left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)\right)\right) \cdot \left(-0.25 \cdot x-scale\right) \]
      *-commutative [=>] 51.2	\[ \left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \cos \left(\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right)\right)\right)\right) \cdot \left(-0.25 \cdot x-scale\right) \]
      associate-*l* [=>] 51.2	\[ \left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \cos \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)\right)\right) \cdot \left(-0.25 \cdot x-scale\right) \]
      *-commutative [=>] 51.2	\[ \left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right) \cdot \color{blue}{\left(x-scale \cdot -0.25\right)} \]

   if -1.75000000000000009e61 < x-scale < -7.1999999999999998e-25

   1. Initial program 62.9

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

   2. Simplified 62.3

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

      [Start] 62.9

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

   3. Taylor expanded in x-scale around inf 60.5

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

   4. Simplified 60.5

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

      [Start] 60.5	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(-1 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)\right)\right) \]
      mul-1-neg [=>] 60.5	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \color{blue}{\left(-\frac{\sqrt{8}}{x-scale \cdot y-scale} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right)}\right) \]
      *-commutative [=>] 60.5	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(-\color{blue}{\sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \cdot \frac{\sqrt{8}}{x-scale \cdot y-scale}}\right)\right) \]
      *-commutative [=>] 60.5	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(-0.25 \cdot \left(-\sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}} \cdot \frac{\sqrt{8}}{\color{blue}{y-scale \cdot x-scale}}\right)\right) \]

   5. Taylor expanded in b around 0 54.0

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

   6. Simplified 54.0

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

      [Start] 54.0	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right) \]
      associate-*r* [=>] 54.0	\[ \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)} \]
      associate-*r* [=>] 54.0	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(\left(\sqrt{2} \cdot a\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 54.0	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\left(\sqrt{2} \cdot a\right) \cdot \color{blue}{\left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \]
      associate-*r* [=>] 54.0	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\left(\sqrt{2} \cdot a\right) \cdot \left(\sqrt{8} \cdot \cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)\right) \]
      *-commutative [=>] 54.0	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\left(\sqrt{2} \cdot a\right) \cdot \left(\sqrt{8} \cdot \cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right) \]

   7. Taylor expanded in a around 0 54.0

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

   8. Simplified 54.0

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

      [Start] 54.0	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right) \]
      *-commutative [=>] 54.0	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \color{blue}{\left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)\right) \]
      associate-*r* [=>] 54.0	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)\right)\right) \]
      *-commutative [<=] 54.0	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right)\right) \]
      *-commutative [=>] 54.0	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right) \]

   if -7.1999999999999998e-25 < x-scale < -3.6499999999999999e-112

   1. Initial program 63.3

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

   2. Simplified 62.7

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

      [Start] 63.3

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

   3. Taylor expanded in y-scale around 0 62.7

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

   4. Taylor expanded in a around -inf 53.8

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

   if -3.6499999999999999e-112 < x-scale < 1.8e-101

   1. Initial program 64.0

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

   2. Simplified 63.7

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

      [Start] 64.0

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

   3. Taylor expanded in angle around 0 49.1

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

   4. Simplified 49.1

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

      [Start] 49.1	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 49.1	\[ \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 49.1	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(b \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]
      *-commutative [=>] 49.1	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left(\sqrt{8} \cdot b\right)} \cdot \sqrt{2}\right) \]

   5. Applied egg-rr 38.0

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

   6. Taylor expanded in y-scale around 0 38.0

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

   7. Simplified 38.0

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

      [Start] 38.0	\[ \left|0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\right| \]
      associate-*r* [=>] 38.0	\[ \left|\color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)}\right| \]
      *-commutative [<=] 38.0	\[ \left|\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{8} \cdot b\right)}\right)\right| \]
      *-commutative [<=] 38.0	\[ \left|\left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(\sqrt{8} \cdot b\right) \cdot \sqrt{2}\right)}\right| \]
      associate-*r* [=>] 38.0	\[ \left|\color{blue}{\left(\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \sqrt{2}}\right| \]
      *-commutative [=>] 38.0	\[ \left|\left(\left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(b \cdot \sqrt{8}\right)}\right) \cdot \sqrt{2}\right| \]
      associate-*l* [<=] 38.0	\[ \left|\color{blue}{\left(\left(\left(0.25 \cdot y-scale\right) \cdot b\right) \cdot \sqrt{8}\right)} \cdot \sqrt{2}\right| \]
      *-commutative [=>] 38.0	\[ \left|\color{blue}{\left(\sqrt{8} \cdot \left(\left(0.25 \cdot y-scale\right) \cdot b\right)\right)} \cdot \sqrt{2}\right| \]
      associate-*l* [=>] 38.0	\[ \left|\color{blue}{\sqrt{8} \cdot \left(\left(\left(0.25 \cdot y-scale\right) \cdot b\right) \cdot \sqrt{2}\right)}\right| \]
      *-commutative [=>] 38.0	\[ \left|\sqrt{8} \cdot \left(\color{blue}{\left(b \cdot \left(0.25 \cdot y-scale\right)\right)} \cdot \sqrt{2}\right)\right| \]

   if 1.8e-101 < x-scale

   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 62.9

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

      [Start] 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 y-scale around 0 46.6

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

   4. Applied egg-rr 35.8

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\mathsf{hypot}\left(\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right) \cdot \sqrt{2}, \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \sqrt{2}\right)}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 41.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.75 \cdot 10^{+61}:\\ \;\;\;\;\left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \left(x-scale \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq -7.2 \cdot 10^{-25}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -3.65 \cdot 10^{-112}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.8 \cdot 10^{-101}:\\ \;\;\;\;\left|\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(0.25 \cdot y-scale\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\sqrt{2} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right), \sqrt{2} \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	39.8
Cost	65732

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := x-scale \cdot \sqrt{8}\\ \mathbf{if}\;x-scale \leq -3 \cdot 10^{-70}:\\ \;\;\;\;\left(-0.25 \cdot \left(\sqrt{2} \cdot t_2\right)\right) \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\sin t_1}^{2}, \left(a \cdot a\right) \cdot {\cos t_1}^{2}\right)}\\ \mathbf{elif}\;x-scale \leq 6.5 \cdot 10^{-101}:\\ \;\;\;\;\left|\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(0.25 \cdot y-scale\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_2 \cdot \mathsf{hypot}\left(\sqrt{2} \cdot \left(a \cdot \cos t_0\right), \sqrt{2} \cdot \left(b \cdot \sin t_0\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	45.8
Cost	39756

\[\begin{array}{l} t_0 := \sqrt{2} \cdot a\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{8} \cdot \left(\left(x-scale \cdot 0.25\right) \cdot t_0\right)\\ \mathbf{elif}\;a \leq 44000000000000:\\ \;\;\;\;\left|\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(0.25 \cdot y-scale\right)\right)\right)\right|\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+119}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(t_0 \cdot \left(\sqrt{8} \cdot \cos \left({\left(\sqrt[3]{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)}^{3}\right)\right)\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+165}:\\ \;\;\;\;\left|0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot b\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \left(x-scale \cdot -0.25\right)\\ \end{array} \]

Alternative 3
Error	45.8
Cost	27024

\[\begin{array}{l} t_0 := \sqrt{2} \cdot a\\ \mathbf{if}\;a \leq -1.55 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{8} \cdot \left(\left(x-scale \cdot 0.25\right) \cdot t_0\right)\\ \mathbf{elif}\;a \leq 20000000000000:\\ \;\;\;\;\left|\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(0.25 \cdot y-scale\right)\right)\right)\right|\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+119}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot t_0\right)\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+168}:\\ \;\;\;\;\left|0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot b\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	45.8
Cost	27024

\[\begin{array}{l} t_0 := \sqrt{2} \cdot a\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{8} \cdot \left(\left(x-scale \cdot 0.25\right) \cdot t_0\right)\\ \mathbf{elif}\;a \leq 4500000000000:\\ \;\;\;\;\left|\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(0.25 \cdot y-scale\right)\right)\right)\right|\\ \mathbf{elif}\;a \leq 10^{+120}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot t_0\right)\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+165}:\\ \;\;\;\;\left|0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot b\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right) \cdot \left(x-scale \cdot -0.25\right)\\ \end{array} \]

Alternative 5
Error	44.8
Cost	20304

\[\begin{array}{l} t_0 := -0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\\ t_1 := \left|0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot b\right)\right)\right)\right|\\ \mathbf{if}\;b \leq -1.14 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-200}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(a \cdot 0.25\right)\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	44.8
Cost	20304

\[\begin{array}{l} t_0 := -0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\\ \mathbf{if}\;b \leq -4.85 \cdot 10^{-150}:\\ \;\;\;\;\left|0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot b\right)\right)\right)\right|\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-200}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(a \cdot 0.25\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(0.25 \cdot y-scale\right)\right)\right)\right|\\ \end{array} \]

Alternative 7
Error	44.9
Cost	20304

\[\begin{array}{l} t_0 := \sqrt{8} \cdot b\\ t_1 := -0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\\ \mathbf{if}\;b \leq -5.7 \cdot 10^{-146}:\\ \;\;\;\;\left|0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot t_0\right)\right)\right|\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-200}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(a \cdot 0.25\right)\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot t_0\right)\right|\\ \end{array} \]

Alternative 8
Error	53.3
Cost	13764

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -1.3 \cdot 10^{-145}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.28 \cdot 10^{-226} \lor \neg \left(x-scale \leq -1.4 \cdot 10^{-304}\right) \land x-scale \leq 3.1 \cdot 10^{-142}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-y-scale\right)\\ \end{array} \]

Alternative 9
Error	52.8
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -4.4 \cdot 10^{-150}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.15 \cdot 10^{-226} \lor \neg \left(x-scale \leq -1.05 \cdot 10^{-298}\right) \land x-scale \leq 1.55 \cdot 10^{-143}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-y-scale\right)\\ \end{array} \]

Alternative 10
Error	52.8
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -4.2 \cdot 10^{-150}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\\ \mathbf{elif}\;x-scale \leq -1.05 \cdot 10^{-226} \lor \neg \left(x-scale \leq -1.08 \cdot 10^{-304}\right) \land x-scale \leq 3.7 \cdot 10^{-141}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-y-scale\right)\\ \end{array} \]

Alternative 11
Error	52.8
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -3.5 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{8} \cdot \left(\left(x-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{elif}\;x-scale \leq -6.5 \cdot 10^{-225} \lor \neg \left(x-scale \leq -4.8 \cdot 10^{-307}\right) \land x-scale \leq 7.5 \cdot 10^{-139}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-y-scale\right)\\ \end{array} \]

Alternative 12
Error	53.6
Cost	653

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -5.3 \cdot 10^{-225} \lor \neg \left(x-scale \leq -2.3 \cdot 10^{-302}\right) \land x-scale \leq 1.75 \cdot 10^{-143}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-y-scale\right)\\ \end{array} \]

Alternative 13
Error	53.9
Cost	192

\[b \cdot y-scale \]

Reproduce

herbie shell --seed 2023011 
(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))))