?

Average Accuracy: 0.9% → 35.1%
Time: 1.9min
Precision: binary64
Cost: 46608

?

\[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
↓
\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\\ t_2 := \cos t_0\\ t_3 := \mathsf{hypot}\left(\sin t_0 \cdot a, t_2 \cdot b\right)\\ t_4 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;y-scale \leq -3 \cdot 10^{-66}:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq -1.25 \cdot 10^{-169}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot t_2\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\mathsf{hypot}\left(a \cdot \sin t_4, b \cdot \cos t_4\right) \cdot \left(\left(y-scale \cdot -0.25\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.5 \cdot 10^{-178}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \left(t_1 \cdot 0.25\right)\\ \end{array} \]
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (*
      (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0)))
      (* (* b a) (* b (- a))))
     (+
      (+
       (/
        (/
         (+
          (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
          (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
         x-scale)
        x-scale)
       (/
        (/
         (+
          (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
          (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
         y-scale)
        y-scale))
      (sqrt
       (+
        (pow
         (-
          (/
           (/
            (+
             (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
             (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
            x-scale)
           x-scale)
          (/
           (/
            (+
             (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
             (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
            y-scale)
           y-scale))
         2.0)
        (pow
         (/
          (/
           (*
            (*
             (* 2.0 (- (pow b 2.0) (pow a 2.0)))
             (sin (* (/ angle 180.0) PI)))
            (cos (* (/ angle 180.0) PI)))
           x-scale)
          y-scale)
         2.0)))))))
  (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))
↓
(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* (sqrt 2.0) (* y-scale (sqrt 8.0))))
        (t_2 (cos t_0))
        (t_3 (hypot (* (sin t_0) a) (* t_2 b)))
        (t_4 (* angle (* 0.005555555555555556 PI))))
   (if (<= y-scale -3e-66)
     (* t_1 (* t_3 -0.25))
     (if (<= y-scale -1.25e-169)
       (* -0.25 (* x-scale (* (sqrt 2.0) (* a (* (sqrt 8.0) t_2)))))
       (if (<= y-scale -1e-283)
         (*
          (hypot (* a (sin t_4)) (* b (cos t_4)))
          (* (* y-scale -0.25) (* (sqrt 2.0) (sqrt 8.0))))
         (if (<= y-scale 3.5e-178)
           (* -0.25 (* (sqrt 2.0) (* x-scale (* (sqrt 8.0) a))))
           (* t_3 (* t_1 0.25))))))))
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0)) / x_45_scale) / x_45_scale) + (((pow((a * cos(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0)) / y_45_scale) / y_45_scale)) + sqrt((pow(((((pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0)) / x_45_scale) / x_45_scale) - (((pow((a * cos(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * sin(((angle / 180.0) * ((double) M_PI)))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(((angle / 180.0) * ((double) M_PI)))) * cos(((angle / 180.0) * ((double) M_PI)))) / x_45_scale) / y_45_scale), 2.0)))))) / ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0));
}
↓
double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sqrt(2.0) * (y_45_scale * sqrt(8.0));
	double t_2 = cos(t_0);
	double t_3 = hypot((sin(t_0) * a), (t_2 * b));
	double t_4 = angle * (0.005555555555555556 * ((double) M_PI));
	double tmp;
	if (y_45_scale <= -3e-66) {
		tmp = t_1 * (t_3 * -0.25);
	} else if (y_45_scale <= -1.25e-169) {
		tmp = -0.25 * (x_45_scale * (sqrt(2.0) * (a * (sqrt(8.0) * t_2))));
	} else if (y_45_scale <= -1e-283) {
		tmp = hypot((a * sin(t_4)), (b * cos(t_4))) * ((y_45_scale * -0.25) * (sqrt(2.0) * sqrt(8.0)));
	} else if (y_45_scale <= 3.5e-178) {
		tmp = -0.25 * (sqrt(2.0) * (x_45_scale * (sqrt(8.0) * a)));
	} else {
		tmp = t_3 * (t_1 * 0.25);
	}
	return tmp;
}

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -Math.sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / Math.pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * Math.PI))), 2.0)) / x_45_scale) / x_45_scale) + (((Math.pow((a * Math.cos(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.sin(((angle / 180.0) * Math.PI))), 2.0)) / y_45_scale) / y_45_scale)) + Math.sqrt((Math.pow(((((Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * Math.PI))), 2.0)) / x_45_scale) / x_45_scale) - (((Math.pow((a * Math.cos(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.sin(((angle / 180.0) * Math.PI))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(((angle / 180.0) * Math.PI))) * Math.cos(((angle / 180.0) * Math.PI))) / x_45_scale) / y_45_scale), 2.0)))))) / ((4.0 * ((b * a) * (b * -a))) / Math.pow((x_45_scale * y_45_scale), 2.0));
}
↓
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sqrt(2.0) * (y_45_scale * Math.sqrt(8.0));
	double t_2 = Math.cos(t_0);
	double t_3 = Math.hypot((Math.sin(t_0) * a), (t_2 * b));
	double t_4 = angle * (0.005555555555555556 * Math.PI);
	double tmp;
	if (y_45_scale <= -3e-66) {
		tmp = t_1 * (t_3 * -0.25);
	} else if (y_45_scale <= -1.25e-169) {
		tmp = -0.25 * (x_45_scale * (Math.sqrt(2.0) * (a * (Math.sqrt(8.0) * t_2))));
	} else if (y_45_scale <= -1e-283) {
		tmp = Math.hypot((a * Math.sin(t_4)), (b * Math.cos(t_4))) * ((y_45_scale * -0.25) * (Math.sqrt(2.0) * Math.sqrt(8.0)));
	} else if (y_45_scale <= 3.5e-178) {
		tmp = -0.25 * (Math.sqrt(2.0) * (x_45_scale * (Math.sqrt(8.0) * a)));
	} else {
		tmp = t_3 * (t_1 * 0.25);
	}
	return tmp;
}

def code(a, b, angle, x_45_scale, y_45_scale):
	return -math.sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / math.pow((x_45_scale * y_45_scale), 2.0))) * ((b * a) * (b * -a))) * (((((math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((angle / 180.0) * math.pi))), 2.0)) / x_45_scale) / x_45_scale) + (((math.pow((a * math.cos(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.sin(((angle / 180.0) * math.pi))), 2.0)) / y_45_scale) / y_45_scale)) + math.sqrt((math.pow(((((math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((angle / 180.0) * math.pi))), 2.0)) / x_45_scale) / x_45_scale) - (((math.pow((a * math.cos(((angle / 180.0) * math.pi))), 2.0) + math.pow((b * math.sin(((angle / 180.0) * math.pi))), 2.0)) / y_45_scale) / y_45_scale)), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(((angle / 180.0) * math.pi))) * math.cos(((angle / 180.0) * math.pi))) / x_45_scale) / y_45_scale), 2.0)))))) / ((4.0 * ((b * a) * (b * -a))) / math.pow((x_45_scale * y_45_scale), 2.0))
↓
def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.sqrt(2.0) * (y_45_scale * math.sqrt(8.0))
	t_2 = math.cos(t_0)
	t_3 = math.hypot((math.sin(t_0) * a), (t_2 * b))
	t_4 = angle * (0.005555555555555556 * math.pi)
	tmp = 0
	if y_45_scale <= -3e-66:
		tmp = t_1 * (t_3 * -0.25)
	elif y_45_scale <= -1.25e-169:
		tmp = -0.25 * (x_45_scale * (math.sqrt(2.0) * (a * (math.sqrt(8.0) * t_2))))
	elif y_45_scale <= -1e-283:
		tmp = math.hypot((a * math.sin(t_4)), (b * math.cos(t_4))) * ((y_45_scale * -0.25) * (math.sqrt(2.0) * math.sqrt(8.0)))
	elif y_45_scale <= 3.5e-178:
		tmp = -0.25 * (math.sqrt(2.0) * (x_45_scale * (math.sqrt(8.0) * a)))
	else:
		tmp = t_3 * (t_1 * 0.25)
	return tmp

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0))) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * Float64(Float64(Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) + Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) + sqrt(Float64((Float64(Float64(Float64(Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) - Float64(Float64(Float64((Float64(a * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(Float64(Float64(angle / 180.0) * pi))) * cos(Float64(Float64(angle / 180.0) * pi))) / x_45_scale) / y_45_scale) ^ 2.0))))))) / Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0)))
end
↓
function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(sqrt(2.0) * Float64(y_45_scale * sqrt(8.0)))
	t_2 = cos(t_0)
	t_3 = hypot(Float64(sin(t_0) * a), Float64(t_2 * b))
	t_4 = Float64(angle * Float64(0.005555555555555556 * pi))
	tmp = 0.0
	if (y_45_scale <= -3e-66)
		tmp = Float64(t_1 * Float64(t_3 * -0.25));
	elseif (y_45_scale <= -1.25e-169)
		tmp = Float64(-0.25 * Float64(x_45_scale * Float64(sqrt(2.0) * Float64(a * Float64(sqrt(8.0) * t_2)))));
	elseif (y_45_scale <= -1e-283)
		tmp = Float64(hypot(Float64(a * sin(t_4)), Float64(b * cos(t_4))) * Float64(Float64(y_45_scale * -0.25) * Float64(sqrt(2.0) * sqrt(8.0))));
	elseif (y_45_scale <= 3.5e-178)
		tmp = Float64(-0.25 * Float64(sqrt(2.0) * Float64(x_45_scale * Float64(sqrt(8.0) * a))));
	else
		tmp = Float64(t_3 * Float64(t_1 * 0.25));
	end
	return tmp
end

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = -sqrt((((2.0 * ((4.0 * ((b * a) * (b * -a))) / ((x_45_scale * y_45_scale) ^ 2.0))) * ((b * a) * (b * -a))) * (((((((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) + (((((a * cos(((angle / 180.0) * pi))) ^ 2.0) + ((b * sin(((angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) + sqrt(((((((((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle / 180.0) * pi))) ^ 2.0)) / x_45_scale) / x_45_scale) - (((((a * cos(((angle / 180.0) * pi))) ^ 2.0) + ((b * sin(((angle / 180.0) * pi))) ^ 2.0)) / y_45_scale) / y_45_scale)) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(((angle / 180.0) * pi))) * cos(((angle / 180.0) * pi))) / x_45_scale) / y_45_scale) ^ 2.0)))))) / ((4.0 * ((b * a) * (b * -a))) / ((x_45_scale * y_45_scale) ^ 2.0));
end
↓
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = sqrt(2.0) * (y_45_scale * sqrt(8.0));
	t_2 = cos(t_0);
	t_3 = hypot((sin(t_0) * a), (t_2 * b));
	t_4 = angle * (0.005555555555555556 * pi);
	tmp = 0.0;
	if (y_45_scale <= -3e-66)
		tmp = t_1 * (t_3 * -0.25);
	elseif (y_45_scale <= -1.25e-169)
		tmp = -0.25 * (x_45_scale * (sqrt(2.0) * (a * (sqrt(8.0) * t_2))));
	elseif (y_45_scale <= -1e-283)
		tmp = hypot((a * sin(t_4)), (b * cos(t_4))) * ((y_45_scale * -0.25) * (sqrt(2.0) * sqrt(8.0)));
	elseif (y_45_scale <= 3.5e-178)
		tmp = -0.25 * (sqrt(2.0) * (x_45_scale * (sqrt(8.0) * a)));
	else
		tmp = t_3 * (t_1 * 0.25);
	end
	tmp_2 = tmp;
end

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[((-N[Sqrt[N[(N[(N[(2.0 * N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] + N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] - N[(N[(N[(N[Power[N[(a * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(y$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Sin[t$95$0], $MachinePrecision] * a), $MachinePrecision] ^ 2 + N[(t$95$2 * b), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -3e-66], N[(t$95$1 * N[(t$95$3 * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, -1.25e-169], N[(-0.25 * N[(x$45$scale * N[(N[Sqrt[2.0], $MachinePrecision] * N[(a * N[(N[Sqrt[8.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, -1e-283], N[(N[Sqrt[N[(a * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(y$45$scale * -0.25), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 3.5e-178], N[(-0.25 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(t$95$1 * 0.25), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}
↓
\begin{array}{l}
t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\
t_1 := \sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\\
t_2 := \cos t_0\\
t_3 := \mathsf{hypot}\left(\sin t_0 \cdot a, t_2 \cdot b\right)\\
t_4 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\
\mathbf{if}\;y-scale \leq -3 \cdot 10^{-66}:\\
\;\;\;\;t_1 \cdot \left(t_3 \cdot -0.25\right)\\

\mathbf{elif}\;y-scale \leq -1.25 \cdot 10^{-169}:\\
\;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot t_2\right)\right)\right)\right)\\

\mathbf{elif}\;y-scale \leq -1 \cdot 10^{-283}:\\
\;\;\;\;\mathsf{hypot}\left(a \cdot \sin t_4, b \cdot \cos t_4\right) \cdot \left(\left(y-scale \cdot -0.25\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\\

\mathbf{elif}\;y-scale \leq 3.5 \cdot 10^{-178}:\\
\;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_3 \cdot \left(t_1 \cdot 0.25\right)\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 5 regimes

  2. if y-scale < -3.0000000000000002e-66

    1. Initial program 1.1%

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

    2. Simplified2.2%

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

      [Start]1.1

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

    3. Taylor expanded in y-scale around inf 6.1%

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

    4. Simplified6.1%

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

      [Start]6.1

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

      associate-*r* [=>]6.1

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

      associate-*r* [=>]6.1

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

      *-commutative [=>]6.1

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

      associate-*r* [<=]6.1

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

      *-commutative [=>]6.1

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

      distribute-lft-out [=>]6.1

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

      +-commutative [=>]6.1

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

    5. Taylor expanded in x-scale around -inf 27.7%

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

    6. Simplified45.3%

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

      [Start]27.7

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

      *-commutative [=>]27.7

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

      associate-*l* [=>]27.7

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

    if -3.0000000000000002e-66 < y-scale < -1.2500000000000001e-169

    1. Initial program 0.6%

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

    2. Simplified1.6%

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

      [Start]0.6

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

    3. Taylor expanded in a around -inf 0.7%

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

    4. Simplified0.7%

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

      [Start]0.7

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

      *-commutative [=>]0.7

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

    5. Taylor expanded in x-scale around inf 17.6%

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

    6. Simplified17.6%

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

      [Start]17.6

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

      *-commutative [=>]17.6

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

    if -1.2500000000000001e-169 < y-scale < -9.99999999999999947e-284

    1. Initial program 0.2%

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

    2. Simplified0.0%

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

      [Start]0.2

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

    3. Taylor expanded in y-scale around inf 3.1%

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

    4. Simplified3.1%

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

      [Start]3.1

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

      associate-*r* [=>]3.1

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

      associate-*r* [=>]3.1

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

      *-commutative [=>]3.1

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

      associate-*r* [<=]3.1

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

      *-commutative [=>]3.1

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

      distribute-lft-out [=>]3.1

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

      +-commutative [=>]3.1

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

    5. Applied egg-rr7.0%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right) \cdot \color{blue}{{\left(2 \cdot {\left(\mathsf{hypot}\left(\frac{a}{\frac{x-scale}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}, \frac{b}{\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right)}^{2}\right)}^{0.5}} \]
      Proof

      [Start]3.1

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

      pow1/2 [=>]3.1

      \[ \left(0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right) \cdot \color{blue}{{\left(2 \cdot \left(\frac{a \cdot a}{\frac{x-scale \cdot x-scale}{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}} + \frac{b \cdot b}{\frac{x-scale \cdot x-scale}{{\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}}\right)\right)}^{0.5}} \]

    6. Simplified6.9%

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

      [Start]7.0

      \[ \left(0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot x-scale\right)\right)\right) \cdot {\left(2 \cdot {\left(\mathsf{hypot}\left(\frac{a}{\frac{x-scale}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}, \frac{b}{\frac{x-scale}{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)\right)}^{2}\right)}^{0.5} \]

      unpow1/2 [=>]7.0

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

      associate-/r/ [=>]6.9

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

      associate-/r/ [=>]6.9

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

    7. Taylor expanded in x-scale around -inf 12.8%

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

    8. Simplified19.7%

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

      [Start]12.8

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

      associate-*r* [=>]12.8

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

      *-commutative [<=]12.8

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

      *-commutative [=>]12.8

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

    if -9.99999999999999947e-284 < y-scale < 3.49999999999999983e-178

    1. Initial program 0.0%

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

    2. Simplified0.0%

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

      [Start]0.0

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

    3. Taylor expanded in a around -inf 0.0%

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

    4. Simplified0.0%

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

      [Start]0.0

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

      *-commutative [=>]0.0

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

    5. Taylor expanded in angle around 0 24.3%

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

    6. Simplified24.3%

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

      [Start]24.3

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

      *-commutative [=>]24.3

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

      associate-*l* [=>]24.3

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

    if 3.49999999999999983e-178 < y-scale

    1. Initial program 1.4%

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

    2. Simplified2.2%

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

      [Start]1.4

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

    3. Taylor expanded in y-scale around inf 5.7%

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

    4. Simplified5.7%

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

      [Start]5.7

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

      associate-*r* [=>]5.7

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

      associate-*r* [=>]5.7

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

      *-commutative [=>]5.7

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

      associate-*r* [<=]5.7

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

      *-commutative [=>]5.7

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

      distribute-lft-out [=>]5.7

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

      +-commutative [=>]5.7

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

    5. Taylor expanded in x-scale around 0 24.7%

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

    6. Simplified40.5%

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

      [Start]24.7

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

      associate-*r* [=>]24.8

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

      *-commutative [<=]24.8

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

      *-commutative [=>]24.8

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

  4. Final simplification35.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -3 \cdot 10^{-66}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right) \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq -1.25 \cdot 10^{-169}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\mathsf{hypot}\left(a \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), b \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(\left(y-scale \cdot -0.25\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.5 \cdot 10^{-178}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right) \cdot \left(\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot 0.25\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy31.6%
Cost46608
\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;y-scale \leq -4 \cdot 10^{-65}:\\ \;\;\;\;{\left({\left(\sqrt[3]{y-scale \cdot b}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;y-scale \leq -6.8 \cdot 10^{-172}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{{\left(\left(b \cdot {16}^{0.5}\right) \cdot \left(y-scale \cdot 0.25\right)\right)}^{2}}\\ \mathbf{elif}\;y-scale \leq 3.4 \cdot 10^{-178}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\mathsf{hypot}\left(a \cdot \sin t_0, b \cdot \cos t_0\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]
Alternative 2
Accuracy34.9%
Cost46608
\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_2 := \cos t_0\\ t_3 := \left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(\sin t_0 \cdot a, t_2 \cdot b\right) \cdot -0.25\right)\\ \mathbf{if}\;y-scale \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y-scale \leq -7.6 \cdot 10^{-171}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot t_2\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -1 \cdot 10^{-283}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y-scale \leq 3.8 \cdot 10^{-181}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\mathsf{hypot}\left(a \cdot \sin t_1, b \cdot \cos t_1\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]
Alternative 3
Accuracy34.9%
Cost46608
\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \sqrt{2} \cdot \sqrt{8}\\ t_3 := \cos t_1\\ t_4 := \mathsf{hypot}\left(a \cdot \sin t_0, b \cdot \cos t_0\right)\\ \mathbf{if}\;y-scale \leq -1.6 \cdot 10^{-65}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(\sin t_1 \cdot a, t_3 \cdot b\right) \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq -5.2 \cdot 10^{-172}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot t_3\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -8.2 \cdot 10^{-284}:\\ \;\;\;\;t_4 \cdot \left(\left(y-scale \cdot -0.25\right) \cdot t_2\right)\\ \mathbf{elif}\;y-scale \leq 1.25 \cdot 10^{-186}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(t_4 \cdot t_2\right)\right)\\ \end{array} \]
Alternative 4
Accuracy28.4%
Cost26760
\[\begin{array}{l} t_0 := {\left({\left(\sqrt[3]{y-scale \cdot b}\right)}^{2}\right)}^{1.5}\\ \mathbf{if}\;y-scale \leq -3.7 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq -2 \cdot 10^{-171}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{{\left(\left(b \cdot {16}^{0.5}\right) \cdot \left(y-scale \cdot 0.25\right)\right)}^{2}}\\ \mathbf{elif}\;y-scale \leq 4 \cdot 10^{-178}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Accuracy28.3%
Cost20172
\[\begin{array}{l} t_0 := -0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ t_1 := {\left({\left(\sqrt[3]{y-scale \cdot b}\right)}^{2}\right)}^{1.5}\\ \mathbf{if}\;y-scale \leq -1.6 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -4 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{{\left(\left(b \cdot {16}^{0.5}\right) \cdot \left(y-scale \cdot 0.25\right)\right)}^{2}}\\ \mathbf{elif}\;y-scale \leq 1.7 \cdot 10^{-179}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Accuracy28.3%
Cost20048
\[\begin{array}{l} t_0 := -0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ t_1 := {\left({\left(\sqrt[3]{y-scale \cdot b}\right)}^{2}\right)}^{1.5}\\ \mathbf{if}\;y-scale \leq -1.65 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -3.3 \cdot 10^{-171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq -1 \cdot 10^{-283}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(y-scale \cdot b\right)}^{2}}\right)}^{1.5}\\ \mathbf{elif}\;y-scale \leq 8.8 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Accuracy20.9%
Cost19785
\[\begin{array}{l} \mathbf{if}\;x-scale \leq -2.3 \cdot 10^{+51} \lor \neg \left(x-scale \leq 3.75 \cdot 10^{-26}\right):\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(y-scale \cdot b\right)}^{2}}\right)}^{1.5}\\ \end{array} \]
Alternative 8
Accuracy19.3%
Cost14433
\[\begin{array}{l} t_0 := -0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ t_1 := y-scale \cdot \left(-b\right)\\ \mathbf{if}\;a \leq -3250000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{b \cdot \left(y-scale \cdot \left(y-scale \cdot b\right)\right)}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-54}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 135000000000:\\ \;\;\;\;\sqrt{y-scale \cdot \left(b \cdot \left(y-scale \cdot b\right)\right)}\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+31} \lor \neg \left(a \leq 7 \cdot 10^{+110}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Accuracy19.3%
Cost14432
\[\begin{array}{l} t_0 := \sqrt{8} \cdot a\\ t_1 := -0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot t_0\right)\right)\\ t_2 := y-scale \cdot \left(-b\right)\\ \mathbf{if}\;a \leq -1300000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{b \cdot \left(y-scale \cdot \left(y-scale \cdot b\right)\right)}\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 240000000000:\\ \;\;\;\;\sqrt{y-scale \cdot \left(b \cdot \left(y-scale \cdot b\right)\right)}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_0\right)\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Accuracy15.9%
Cost7236
\[\begin{array}{l} \mathbf{if}\;angle \leq -4.4 \cdot 10^{-189}:\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(y-scale \cdot 0.25\right) \cdot \left(\left(y-scale \cdot b\right) \cdot \left(b \cdot 4\right)\right)}\\ \end{array} \]
Alternative 11
Accuracy17.1%
Cost6980
\[\begin{array}{l} \mathbf{if}\;y-scale \leq -4 \cdot 10^{+146}:\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{b \cdot \left(y-scale \cdot \left(y-scale \cdot b\right)\right)}\\ \end{array} \]
Alternative 12
Accuracy15.9%
Cost6980
\[\begin{array}{l} \mathbf{if}\;angle \leq -3.05 \cdot 10^{-192}:\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y-scale \cdot \left(b \cdot \left(y-scale \cdot b\right)\right)}\\ \end{array} \]
Alternative 13
Accuracy15.8%
Cost786
\[\begin{array}{l} \mathbf{if}\;angle \leq -5.1 \cdot 10^{-140} \lor \neg \left(angle \leq -2.3 \cdot 10^{-286} \lor \neg \left(angle \leq 6.2 \cdot 10^{-241}\right) \land angle \leq 1.45 \cdot 10^{-134}\right):\\ \;\;\;\;y-scale \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
Alternative 14
Accuracy16.4%
Cost192
\[y-scale \cdot b \]

Error

Reproduce?

herbie shell --seed 2023151 
(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))))