a from scale-rotated-ellipse

Average Error: 63.4 → 49.5

Time: 1.7min

Precision: binary64

Cost: 72260

Math

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

↓

\[\begin{array}{l} t_0 := x-scale \cdot \sqrt{8}\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := {\sin t_1}^{2}\\ t_3 := \cos t_1\\ t_4 := {a}^{2} \cdot {t_3}^{2}\\ \mathbf{if}\;x-scale \leq -3.4 \cdot 10^{+42}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\left(\sqrt{2} \cdot t_0\right) \cdot \sqrt{t_4 + {b}^{2} \cdot t_2}\right)\right)\\ \mathbf{elif}\;x-scale \leq -2.65 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \frac{\sqrt{2} \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot b\right)\right)\right)}{1}\right)\\ \mathbf{elif}\;x-scale \leq -3.2 \cdot 10^{-109}:\\ \;\;\;\;0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -4 \cdot 10^{-252}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(t_3 \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.4 \cdot 10^{+107}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \sqrt{2 \cdot \left(t_4 + t_2 \cdot {b}^{2}\right)}\right)\\ \end{array} \]

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 (* angle PI)))
        (t_2 (pow (sin t_1) 2.0))
        (t_3 (cos t_1))
        (t_4 (* (pow a 2.0) (pow t_3 2.0))))
   (if (<= x-scale -3.4e+42)
     (*
      0.25
      (* -1.0 (* (* (sqrt 2.0) t_0) (sqrt (+ t_4 (* (pow b 2.0) t_2))))))
     (if (<= x-scale -2.65e-27)
       (*
        0.25
        (*
         -1.0
         (/
          (*
           (sqrt 2.0)
           (*
            (cos (* angle (* 0.005555555555555556 PI)))
            (* (sqrt 8.0) (* y-scale b))))
          1.0)))
       (if (<= x-scale -3.2e-109)
         (*
          0.25
          (-
           (*
            (sqrt 2.0)
            (*
             (cos (* PI (* 0.005555555555555556 angle)))
             (* (sqrt 8.0) (* x-scale a))))))
         (if (<= x-scale -4e-252)
           (* 0.25 (* y-scale (* (sqrt 2.0) (* b (* t_3 (sqrt 8.0))))))
           (if (<= x-scale 3.4e+107)
             (* 0.25 (* -1.0 (* (sqrt 2.0) (* y-scale (* b (sqrt 8.0))))))
             (* 0.25 (* t_0 (sqrt (* 2.0 (+ t_4 (* t_2 (pow b 2.0))))))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0)) / x_45_scale) / x_45_scale) + (((pow((a * cos(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0)) / y_45_scale) / y_45_scale)) + sqrt((pow(((((pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0)) / x_45_scale) / x_45_scale) - (((pow((a * cos(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(((angle / 180.0) * ((double) M_PI)))) * cos(((angle / 180.0) * ((double) M_PI)))) / x_45_scale) / y_45_scale), 2.0)))))) / ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = x_45_scale * sqrt(8.0);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = pow(sin(t_1), 2.0);
	double t_3 = cos(t_1);
	double t_4 = pow(a, 2.0) * pow(t_3, 2.0);
	double tmp;
	if (x_45_scale <= -3.4e+42) {
		tmp = 0.25 * (-1.0 * ((sqrt(2.0) * t_0) * sqrt((t_4 + (pow(b, 2.0) * t_2)))));
	} else if (x_45_scale <= -2.65e-27) {
		tmp = 0.25 * (-1.0 * ((sqrt(2.0) * (cos((angle * (0.005555555555555556 * ((double) M_PI)))) * (sqrt(8.0) * (y_45_scale * b)))) / 1.0));
	} else if (x_45_scale <= -3.2e-109) {
		tmp = 0.25 * -(sqrt(2.0) * (cos((((double) M_PI) * (0.005555555555555556 * angle))) * (sqrt(8.0) * (x_45_scale * a))));
	} else if (x_45_scale <= -4e-252) {
		tmp = 0.25 * (y_45_scale * (sqrt(2.0) * (b * (t_3 * sqrt(8.0)))));
	} else if (x_45_scale <= 3.4e+107) {
		tmp = 0.25 * (-1.0 * (sqrt(2.0) * (y_45_scale * (b * sqrt(8.0)))));
	} else {
		tmp = 0.25 * (t_0 * sqrt((2.0 * (t_4 + (t_2 * pow(b, 2.0))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -Math.sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / Math.pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * Math.PI))), 2.0)) / x_45_scale) / x_45_scale) + (((Math.pow((a * Math.cos(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.sin(((angle / 180.0) * Math.PI))), 2.0)) / y_45_scale) / y_45_scale)) + Math.sqrt((Math.pow(((((Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * Math.PI))), 2.0)) / x_45_scale) / x_45_scale) - (((Math.pow((a * Math.cos(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.sin(((angle / 180.0) * Math.PI))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(((angle / 180.0) * Math.PI))) * Math.cos(((angle / 180.0) * Math.PI))) / x_45_scale) / y_45_scale), 2.0)))))) / ((4.0 * ((b * a) * (b * -a))) / Math.pow((x_45_scale * y_45_scale), 2.0));
}

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = x_45_scale * Math.sqrt(8.0);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.pow(Math.sin(t_1), 2.0);
	double t_3 = Math.cos(t_1);
	double t_4 = Math.pow(a, 2.0) * Math.pow(t_3, 2.0);
	double tmp;
	if (x_45_scale <= -3.4e+42) {
		tmp = 0.25 * (-1.0 * ((Math.sqrt(2.0) * t_0) * Math.sqrt((t_4 + (Math.pow(b, 2.0) * t_2)))));
	} else if (x_45_scale <= -2.65e-27) {
		tmp = 0.25 * (-1.0 * ((Math.sqrt(2.0) * (Math.cos((angle * (0.005555555555555556 * Math.PI))) * (Math.sqrt(8.0) * (y_45_scale * b)))) / 1.0));
	} else if (x_45_scale <= -3.2e-109) {
		tmp = 0.25 * -(Math.sqrt(2.0) * (Math.cos((Math.PI * (0.005555555555555556 * angle))) * (Math.sqrt(8.0) * (x_45_scale * a))));
	} else if (x_45_scale <= -4e-252) {
		tmp = 0.25 * (y_45_scale * (Math.sqrt(2.0) * (b * (t_3 * Math.sqrt(8.0)))));
	} else if (x_45_scale <= 3.4e+107) {
		tmp = 0.25 * (-1.0 * (Math.sqrt(2.0) * (y_45_scale * (b * Math.sqrt(8.0)))));
	} else {
		tmp = 0.25 * (t_0 * Math.sqrt((2.0 * (t_4 + (t_2 * Math.pow(b, 2.0))))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return -math.sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / math.pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((angle / 180.0) * math.pi))), 2.0)) / x_45_scale) / x_45_scale) + (((math.pow((a * math.cos(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.sin(((angle / 180.0) * math.pi))), 2.0)) / y_45_scale) / y_45_scale)) + math.sqrt((math.pow(((((math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((angle / 180.0) * math.pi))), 2.0)) / x_45_scale) / x_45_scale) - (((math.pow((a * math.cos(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.sin(((angle / 180.0) * math.pi))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(((angle / 180.0) * math.pi))) * math.cos(((angle / 180.0) * math.pi))) / x_45_scale) / y_45_scale), 2.0)))))) / ((4.0 * ((b * a) * (b * -a))) / math.pow((x_45_scale * y_45_scale), 2.0))

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = x_45_scale * math.sqrt(8.0)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.pow(math.sin(t_1), 2.0)
	t_3 = math.cos(t_1)
	t_4 = math.pow(a, 2.0) * math.pow(t_3, 2.0)
	tmp = 0
	if x_45_scale <= -3.4e+42:
		tmp = 0.25 * (-1.0 * ((math.sqrt(2.0) * t_0) * math.sqrt((t_4 + (math.pow(b, 2.0) * t_2)))))
	elif x_45_scale <= -2.65e-27:
		tmp = 0.25 * (-1.0 * ((math.sqrt(2.0) * (math.cos((angle * (0.005555555555555556 * math.pi))) * (math.sqrt(8.0) * (y_45_scale * b)))) / 1.0))
	elif x_45_scale <= -3.2e-109:
		tmp = 0.25 * -(math.sqrt(2.0) * (math.cos((math.pi * (0.005555555555555556 * angle))) * (math.sqrt(8.0) * (x_45_scale * a))))
	elif x_45_scale <= -4e-252:
		tmp = 0.25 * (y_45_scale * (math.sqrt(2.0) * (b * (t_3 * math.sqrt(8.0)))))
	elif x_45_scale <= 3.4e+107:
		tmp = 0.25 * (-1.0 * (math.sqrt(2.0) * (y_45_scale * (b * math.sqrt(8.0)))))
	else:
		tmp = 0.25 * (t_0 * math.sqrt((2.0 * (t_4 + (t_2 * math.pow(b, 2.0))))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0))) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * Float64(Float64(Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) + Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) + sqrt(Float64((Float64(Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) - Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(Float64(angle / 180.0) * pi))) * cos(Float64(Float64(angle / 180.0) * pi))) / x_45_scale) / y_45_scale) ^ 2.0))))))) / Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0)))
end

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(x_45_scale * sqrt(8.0))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = sin(t_1) ^ 2.0
	t_3 = cos(t_1)
	t_4 = Float64((a ^ 2.0) * (t_3 ^ 2.0))
	tmp = 0.0
	if (x_45_scale <= -3.4e+42)
		tmp = Float64(0.25 * Float64(-1.0 * Float64(Float64(sqrt(2.0) * t_0) * sqrt(Float64(t_4 + Float64((b ^ 2.0) * t_2))))));
	elseif (x_45_scale <= -2.65e-27)
		tmp = Float64(0.25 * Float64(-1.0 * Float64(Float64(sqrt(2.0) * Float64(cos(Float64(angle * Float64(0.005555555555555556 * pi))) * Float64(sqrt(8.0) * Float64(y_45_scale * b)))) / 1.0)));
	elseif (x_45_scale <= -3.2e-109)
		tmp = Float64(0.25 * Float64(-Float64(sqrt(2.0) * Float64(cos(Float64(pi * Float64(0.005555555555555556 * angle))) * Float64(sqrt(8.0) * Float64(x_45_scale * a))))));
	elseif (x_45_scale <= -4e-252)
		tmp = Float64(0.25 * Float64(y_45_scale * Float64(sqrt(2.0) * Float64(b * Float64(t_3 * sqrt(8.0))))));
	elseif (x_45_scale <= 3.4e+107)
		tmp = Float64(0.25 * Float64(-1.0 * Float64(sqrt(2.0) * Float64(y_45_scale * Float64(b * sqrt(8.0))))));
	else
		tmp = Float64(0.25 * Float64(t_0 * sqrt(Float64(2.0 * Float64(t_4 + Float64(t_2 * (b ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = -sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / ((x_45_scale * y_45_scale) ^ 2.0))) * ((b * a) * (b * -a))) * (((((((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) + (((((a * cos(((angle / 180.0) * pi))) ^ 2.0) + ((b * sin(((angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) + sqrt(((((((((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) - (((((a * cos(((angle / 180.0) * pi))) ^ 2.0) + ((b * sin(((angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(((angle / 180.0) * pi))) * cos(((angle / 180.0) * pi))) / x_45_scale) / y_45_scale) ^ 2.0)))))) / ((4.0 * ((b * a) * (b * -a))) / ((x_45_scale * y_45_scale) ^ 2.0));
end

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = x_45_scale * sqrt(8.0);
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = sin(t_1) ^ 2.0;
	t_3 = cos(t_1);
	t_4 = (a ^ 2.0) * (t_3 ^ 2.0);
	tmp = 0.0;
	if (x_45_scale <= -3.4e+42)
		tmp = 0.25 * (-1.0 * ((sqrt(2.0) * t_0) * sqrt((t_4 + ((b ^ 2.0) * t_2)))));
	elseif (x_45_scale <= -2.65e-27)
		tmp = 0.25 * (-1.0 * ((sqrt(2.0) * (cos((angle * (0.005555555555555556 * pi))) * (sqrt(8.0) * (y_45_scale * b)))) / 1.0));
	elseif (x_45_scale <= -3.2e-109)
		tmp = 0.25 * -(sqrt(2.0) * (cos((pi * (0.005555555555555556 * angle))) * (sqrt(8.0) * (x_45_scale * a))));
	elseif (x_45_scale <= -4e-252)
		tmp = 0.25 * (y_45_scale * (sqrt(2.0) * (b * (t_3 * sqrt(8.0)))));
	elseif (x_45_scale <= 3.4e+107)
		tmp = 0.25 * (-1.0 * (sqrt(2.0) * (y_45_scale * (b * sqrt(8.0)))));
	else
		tmp = 0.25 * (t_0 * sqrt((2.0 * (t_4 + (t_2 * (b ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[((-N[Sqrt[N[(N[(N[(2.0 * N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] + N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] - N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(x$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -3.4e+42], N[(0.25 * N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(t$95$4 + N[(N[Power[b, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -2.65e-27], N[(0.25 * N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(y$45$scale * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -3.2e-109], N[(0.25 * (-N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(x$45$scale * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[x$45$scale, -4e-252], N[(0.25 * N[(y$45$scale * N[(N[Sqrt[2.0], $MachinePrecision] * N[(b * N[(t$95$3 * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 3.4e+107], N[(0.25 * N[(-1.0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(y$45$scale * N[(b * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(t$95$0 * N[Sqrt[N[(2.0 * N[(t$95$4 + N[(t$95$2 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x-scale < -3.39999999999999975e42

   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 63.2

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

      [Start] 63.3

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

   3. Taylor expanded in x-scale around inf 58.8

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

   4. Simplified 58.8

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

      [Start] 58.8	\[ 0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 58.8	\[ 0.25 \cdot \left(\color{blue}{\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 58.8	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} \cdot 2} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 58.8	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}}\right) \]

   5. Taylor expanded in y-scale around -inf 42.5

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

   if -3.39999999999999975e42 < x-scale < -2.65000000000000003e-27

   1. Initial program 62.7

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

   2. Simplified 62.7

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

      [Start] 62.7

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

   3. Taylor expanded in y-scale around inf 59.0

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

   4. Simplified 59.0

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

      [Start] 59.0	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 59.0	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 59.0	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{\color{blue}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}}\right) \]

   5. Taylor expanded in b around -inf 56.0

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

   6. Applied egg-rr 56.0

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

   if -2.65000000000000003e-27 < x-scale < -3.2000000000000002e-109

   1. Initial program 63.2

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

   2. Simplified 63.0

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

      [Start] 63.2

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

   3. Taylor expanded in x-scale around inf 61.9

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

   4. Simplified 61.9

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

      [Start] 61.9	\[ 0.25 \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 61.9	\[ 0.25 \cdot \left(\color{blue}{\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 61.9	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} \cdot 2} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 61.9	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}}\right) \]

   5. Taylor expanded in a around -inf 55.5

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

   6. Simplified 55.4

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

      [Start] 55.5	\[ 0.25 \cdot \left(-1 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 55.5	\[ 0.25 \cdot \left(-1 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 55.5	\[ 0.25 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right) \cdot -1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 55.5	\[ 0.25 \cdot \color{blue}{\left(-\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 55.5	\[ 0.25 \cdot \left(-\sqrt{2} \cdot \left(x-scale \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a \cdot \sqrt{8}\right)\right)}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 55.5	\[ 0.25 \cdot \left(-\sqrt{2} \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 55.5	\[ 0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 55.4	\[ 0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 55.4	\[ 0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(x-scale \cdot \color{blue}{\left(\sqrt{8} \cdot a\right)}\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 55.4	\[ 0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \color{blue}{\left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)}\right)\right) \]

   if -3.2000000000000002e-109 < x-scale < -3.99999999999999977e-252

   1. Initial program 63.9

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

   2. Simplified 63.8

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

      [Start] 63.9

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

   3. Taylor expanded in y-scale around inf 61.6

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

   4. Simplified 61.6

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

      [Start] 61.6	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 61.6	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 61.6	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{\color{blue}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}}\right) \]

   5. Taylor expanded in a around 0 49.5

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

   if -3.99999999999999977e-252 < x-scale < 3.3999999999999997e107

   1. Initial program 63.4

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

   2. Simplified 63.4

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

      [Start] 63.4

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

   3. Taylor expanded in y-scale around inf 60.9

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

   4. Simplified 60.9

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

      [Start] 60.9	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 60.9	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 60.9	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + 2 \cdot \frac{\color{blue}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}}\right) \]

   5. Taylor expanded in b around -inf 52.5

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

   6. Taylor expanded in angle around 0 52.5

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

   if 3.3999999999999997e107 < x-scale

   1. Initial program 63.7

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

   2. Simplified 63.6

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

      [Start] 63.7

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

   3. Taylor expanded in y-scale around 0 40.4

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

   4. Simplified 40.4

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

      [Start] 40.4	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 40.4	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{\left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) \cdot 2} + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 40.4	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 40.4	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + \color{blue}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}}\right)}\right) \]

3. Recombined 6 regimes into one program.

4. Final simplification 49.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -3.4 \cdot 10^{+42}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}\right)\right)\\ \mathbf{elif}\;x-scale \leq -2.65 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \frac{\sqrt{2} \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot b\right)\right)\right)}{1}\right)\\ \mathbf{elif}\;x-scale \leq -3.2 \cdot 10^{-109}:\\ \;\;\;\;0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -4 \cdot 10^{-252}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.4 \cdot 10^{+107}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	48.4
Cost	72792

\[\begin{array}{l} t_0 := x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\\ t_1 := 0.25 \cdot t_0\\ t_2 := y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\\ t_3 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_4 := {a}^{2} \cdot {\sin t_3}^{2}\\ \mathbf{if}\;y-scale \leq -3.5 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(-t_2 \cdot \sqrt{1 \cdot {b}^{2} + t_4}\right)\\ \mathbf{elif}\;y-scale \leq -1.8 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 4.5 \cdot 10^{-307}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot t_0\right)\\ \mathbf{elif}\;y-scale \leq 5.3 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 8.6 \cdot 10^{-109}:\\ \;\;\;\;0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.6 \cdot 10^{+32}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_2 \cdot \sqrt{{\cos t_3}^{2} \cdot {b}^{2} + t_4}\right)\\ \end{array} \]

Alternative 2
Error	48.3
Cost	72792

\[\begin{array}{l} t_0 := x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\\ t_1 := 0.25 \cdot t_0\\ t_2 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_3 := {a}^{2} \cdot {\sin t_2}^{2}\\ \mathbf{if}\;y-scale \leq -1.85 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(-\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \sqrt{1 \cdot {b}^{2} + t_3}\right)\\ \mathbf{elif}\;y-scale \leq -1 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -2.8 \cdot 10^{-307}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot t_0\right)\\ \mathbf{elif}\;y-scale \leq 8.6 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 2.3 \cdot 10^{-107}:\\ \;\;\;\;0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.1 \cdot 10^{+30}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{{\cos t_2}^{2} \cdot {b}^{2} + t_3}\right)\\ \end{array} \]

Alternative 3
Error	49.5
Cost	72196

\[\begin{array}{l} t_0 := x-scale \cdot \sqrt{8}\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := \cos t_1\\ t_3 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_4 := \cos t_3\\ \mathbf{if}\;x-scale \leq -2.4 \cdot 10^{+42}:\\ \;\;\;\;0.25 \cdot \left(-\sqrt{2} \cdot \left(\sqrt{{a}^{2} \cdot {t_2}^{2} + {b}^{2} \cdot {\sin t_1}^{2}} \cdot t_0\right)\right)\\ \mathbf{elif}\;x-scale \leq -2.5 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \frac{\sqrt{2} \cdot \left(\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot b\right)\right)\right)}{1}\right)\\ \mathbf{elif}\;x-scale \leq -9.5 \cdot 10^{-103}:\\ \;\;\;\;0.25 \cdot \left(-\sqrt{2} \cdot \left(t_2 \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq -6.2 \cdot 10^{-252}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(t_4 \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3 \cdot 10^{+108}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {t_4}^{2} + {\sin t_3}^{2} \cdot {b}^{2}\right)}\right)\\ \end{array} \]

Alternative 4
Error	49.5
Cost	52740

\[\begin{array}{l} t_0 := x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\\ t_1 := 0.25 \cdot \left(-1 \cdot t_0\right)\\ t_2 := 0.25 \cdot t_0\\ \mathbf{if}\;y-scale \leq -7.2 \cdot 10^{-96}:\\ \;\;\;\;0.25 \cdot \left(-\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \sqrt{1 \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}}\right)\\ \mathbf{elif}\;y-scale \leq -1.3 \cdot 10^{-209}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq -3.8 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 8.6 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 7.5 \cdot 10^{-110}:\\ \;\;\;\;0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3700000000:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.1 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{8} \cdot y-scale\right) \cdot \left(0.25 \cdot \sqrt{1 \cdot \left({b}^{2} + {b}^{2}\right)}\right)\\ \end{array} \]

Alternative 5
Error	51.8
Cost	27548

\[\begin{array}{l} t_0 := x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\\ t_1 := 0.25 \cdot \left(-1 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot b\right)\right)\right)\right)\\ t_2 := 0.25 \cdot t_0\\ t_3 := 0.25 \cdot \left(-1 \cdot t_0\right)\\ \mathbf{if}\;y-scale \leq -7.6 \cdot 10^{-31}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(b \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -2 \cdot 10^{-210}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y-scale \leq 1.2 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 7.5 \cdot 10^{-108}:\\ \;\;\;\;0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3200000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 2.3 \cdot 10^{+123}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	51.2
Cost	27548

\[\begin{array}{l} t_0 := x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\\ t_1 := 0.25 \cdot t_0\\ t_2 := 0.25 \cdot \left(-1 \cdot t_0\right)\\ \mathbf{if}\;y-scale \leq -6.5 \cdot 10^{-32}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(b \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -7.5 \cdot 10^{-215}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 6 \cdot 10^{-307}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 1.25 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 4.4 \cdot 10^{-110}:\\ \;\;\;\;0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 4400000000:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.4 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{8} \cdot y-scale\right) \cdot \left(0.25 \cdot \sqrt{1 \cdot \left({b}^{2} + {b}^{2}\right)}\right)\\ \end{array} \]

Alternative 7
Error	51.8
Cost	27484

\[\begin{array}{l} t_0 := 0.25 \cdot \left(-1 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot b\right)\right)\right)\right)\\ t_1 := 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)\\ t_2 := 0.25 \cdot \left(-\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \mathbf{if}\;y-scale \leq -6.5 \cdot 10^{-30}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(b \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -1.3 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -2.9 \cdot 10^{-307}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 1.3 \cdot 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq 2.9 \cdot 10^{-108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y-scale \leq 600000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 5.2 \cdot 10^{+124}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	52.1
Cost	26760

\[\begin{array}{l} t_0 := 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{if}\;x-scale \leq -1.7 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 2.6 \cdot 10^{+94}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	52.1
Cost	26760

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -2.1 \cdot 10^{+91}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	52.1
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -3.7 \cdot 10^{+89}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 2.25 \cdot 10^{+94}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(b \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	52.1
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -2.1 \cdot 10^{+91}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 4.6 \cdot 10^{+97}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	52.1
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -5.8 \cdot 10^{+89}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 4.7 \cdot 10^{+95}:\\ \;\;\;\;0.25 \cdot \left(-1 \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	52.1
Cost	13704

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -3.1 \cdot 10^{+90}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 4.4 \cdot 10^{+96}:\\ \;\;\;\;0.25 \cdot \left(-\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]

Alternative 14
Error	51.1
Cost	13640

\[\begin{array}{l} t_0 := 0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{if}\;x-scale \leq -2.2 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 4.3 \cdot 10^{+80}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	51.1
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -3.4 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 2 \cdot 10^{+85}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]

Alternative 16
Error	51.1
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -1.8 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	53.7
Cost	13376

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

Alternative 18
Error	53.6
Cost	13376

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

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))))