b from scale-rotated-ellipse

Average Error: 64.0 → 40.8

Time: 2.5min

Precision: binary64

Cost: 13840

Math

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

↓

\[\begin{array}{l} t_0 := x-scale \cdot \left(-a\right)\\ \mathbf{if}\;y-scale \leq -1.5 \cdot 10^{+117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq -1.25 \cdot 10^{-51}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;y-scale \leq 10^{-184}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot 0\right)\right)\\ \mathbf{elif}\;y-scale \leq 6.2 \cdot 10^{+32}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \mathsf{hypot}\left(b, b\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 7 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 3.3 \cdot 10^{+134}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]

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 (- a))))
   (if (<= y-scale -1.5e+117)
     t_0
     (if (<= y-scale -1.25e-51)
       (* x-scale a)
       (if (<= y-scale 1e-184)
         (* 0.25 (* y-scale (* (sqrt 8.0) 0.0)))
         (if (<= y-scale 6.2e+32)
           (* 0.25 (* y-scale (* (sqrt 8.0) (hypot b b))))
           (if (<= y-scale 7e+122)
             t_0
             (if (<= y-scale 3.3e+134)
               (* 0.25 (* y-scale (sqrt (* (* b b) 16.0))))
               (* x-scale a)))))))))

C

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

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = x_45_scale * -a;
	double tmp;
	if (y_45_scale <= -1.5e+117) {
		tmp = t_0;
	} else if (y_45_scale <= -1.25e-51) {
		tmp = x_45_scale * a;
	} else if (y_45_scale <= 1e-184) {
		tmp = 0.25 * (y_45_scale * (sqrt(8.0) * 0.0));
	} else if (y_45_scale <= 6.2e+32) {
		tmp = 0.25 * (y_45_scale * (sqrt(8.0) * hypot(b, b)));
	} else if (y_45_scale <= 7e+122) {
		tmp = t_0;
	} else if (y_45_scale <= 3.3e+134) {
		tmp = 0.25 * (y_45_scale * sqrt(((b * b) * 16.0)));
	} else {
		tmp = x_45_scale * a;
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = x_45_scale * -a;
	double tmp;
	if (y_45_scale <= -1.5e+117) {
		tmp = t_0;
	} else if (y_45_scale <= -1.25e-51) {
		tmp = x_45_scale * a;
	} else if (y_45_scale <= 1e-184) {
		tmp = 0.25 * (y_45_scale * (Math.sqrt(8.0) * 0.0));
	} else if (y_45_scale <= 6.2e+32) {
		tmp = 0.25 * (y_45_scale * (Math.sqrt(8.0) * Math.hypot(b, b)));
	} else if (y_45_scale <= 7e+122) {
		tmp = t_0;
	} else if (y_45_scale <= 3.3e+134) {
		tmp = 0.25 * (y_45_scale * Math.sqrt(((b * b) * 16.0)));
	} else {
		tmp = x_45_scale * a;
	}
	return tmp;
}

Python

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

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = x_45_scale * -a
	tmp = 0
	if y_45_scale <= -1.5e+117:
		tmp = t_0
	elif y_45_scale <= -1.25e-51:
		tmp = x_45_scale * a
	elif y_45_scale <= 1e-184:
		tmp = 0.25 * (y_45_scale * (math.sqrt(8.0) * 0.0))
	elif y_45_scale <= 6.2e+32:
		tmp = 0.25 * (y_45_scale * (math.sqrt(8.0) * math.hypot(b, b)))
	elif y_45_scale <= 7e+122:
		tmp = t_0
	elif y_45_scale <= 3.3e+134:
		tmp = 0.25 * (y_45_scale * math.sqrt(((b * b) * 16.0)))
	else:
		tmp = x_45_scale * a
	return tmp

Julia

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

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(x_45_scale * Float64(-a))
	tmp = 0.0
	if (y_45_scale <= -1.5e+117)
		tmp = t_0;
	elseif (y_45_scale <= -1.25e-51)
		tmp = Float64(x_45_scale * a);
	elseif (y_45_scale <= 1e-184)
		tmp = Float64(0.25 * Float64(y_45_scale * Float64(sqrt(8.0) * 0.0)));
	elseif (y_45_scale <= 6.2e+32)
		tmp = Float64(0.25 * Float64(y_45_scale * Float64(sqrt(8.0) * hypot(b, b))));
	elseif (y_45_scale <= 7e+122)
		tmp = t_0;
	elseif (y_45_scale <= 3.3e+134)
		tmp = Float64(0.25 * Float64(y_45_scale * sqrt(Float64(Float64(b * b) * 16.0))));
	else
		tmp = Float64(x_45_scale * a);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = x_45_scale * -a;
	tmp = 0.0;
	if (y_45_scale <= -1.5e+117)
		tmp = t_0;
	elseif (y_45_scale <= -1.25e-51)
		tmp = x_45_scale * a;
	elseif (y_45_scale <= 1e-184)
		tmp = 0.25 * (y_45_scale * (sqrt(8.0) * 0.0));
	elseif (y_45_scale <= 6.2e+32)
		tmp = 0.25 * (y_45_scale * (sqrt(8.0) * hypot(b, b)));
	elseif (y_45_scale <= 7e+122)
		tmp = t_0;
	elseif (y_45_scale <= 3.3e+134)
		tmp = 0.25 * (y_45_scale * sqrt(((b * b) * 16.0)));
	else
		tmp = x_45_scale * a;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(x$45$scale * (-a)), $MachinePrecision]}, If[LessEqual[y$45$scale, -1.5e+117], t$95$0, If[LessEqual[y$45$scale, -1.25e-51], N[(x$45$scale * a), $MachinePrecision], If[LessEqual[y$45$scale, 1e-184], N[(0.25 * N[(y$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 6.2e+32], N[(0.25 * N[(y$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[b ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 7e+122], t$95$0, If[LessEqual[y$45$scale, 3.3e+134], N[(0.25 * N[(y$45$scale * N[Sqrt[N[(N[(b * b), $MachinePrecision] * 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$45$scale * a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y-scale < -1.5e117 or 6.19999999999999986e32 < y-scale < 7.00000000000000028e122

   1. Initial program 64.0

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

   2. Taylor expanded in angle around 0 60.8

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 60.8

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

      [Start] 60.8	\[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
      associate-*r/ [=>] 60.8	\[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\frac{2 \cdot {a}^{2}}{{y-scale}^{2}}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
      unpow2 [=>] 60.8	\[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{2 \cdot \color{blue}{\left(a \cdot a\right)}}{{y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
      unpow2 [=>] 60.8	\[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{2 \cdot \left(a \cdot a\right)}{\color{blue}{y-scale \cdot y-scale}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   4. Taylor expanded in a around -inf 47.2

      \[\leadsto \color{blue}{-1 \cdot \left(x-scale \cdot a\right)} \]

   5. Simplified 47.2

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

      [Start] 47.2	\[ -1 \cdot \left(x-scale \cdot a\right) \]
      mul-1-neg [=>] 47.2	\[ \color{blue}{-x-scale \cdot a} \]
      distribute-rgt-neg-in [=>] 47.2	\[ \color{blue}{x-scale \cdot \left(-a\right)} \]

   if -1.5e117 < y-scale < -1.25000000000000001e-51 or 3.3e134 < y-scale

   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. Taylor expanded in angle around 0 61.7

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 61.7

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

      [Start] 61.7	\[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
      associate-*r/ [=>] 61.7	\[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\frac{2 \cdot {a}^{2}}{{y-scale}^{2}}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
      unpow2 [=>] 61.7	\[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{2 \cdot \color{blue}{\left(a \cdot a\right)}}{{y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
      unpow2 [=>] 61.7	\[ \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{2 \cdot \left(a \cdot a\right)}{\color{blue}{y-scale \cdot y-scale}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   4. Taylor expanded in b around 0 47.0

      \[\leadsto \color{blue}{x-scale \cdot a} \]

   if -1.25000000000000001e-51 < y-scale < 1.0000000000000001e-184

   1. Initial program 64.0

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

   2. Simplified 63.5

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

      [Start] 64.0

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

   3. Taylor expanded in x-scale around inf 63.2

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - 0.5 \cdot \frac{-2 \cdot \left(\left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) \cdot \left(\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right) + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\left({b}^{2} - {a}^{2}\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}{\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right)} \]

   4. Simplified 63.2

      \[\leadsto \color{blue}{0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right) - 0.5 \cdot \frac{\mathsf{fma}\left(-2, \left(\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right) \cdot \mathsf{fma}\left(b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right), \frac{4 \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\left(b \cdot b - a \cdot a\right)}^{2}}}\right)}{\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}}}\right)\right)} \]
      Proof

      [Start] 63.2

      \[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - 0.5 \cdot \frac{-2 \cdot \left(\left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) \cdot \left(\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right) + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\left({b}^{2} - {a}^{2}\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}{\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right) \]

   5. Taylor expanded in angle around 0 40.8

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

   6. Simplified 40.8

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

      [Start] 40.8	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{{b}^{2} - -1 \cdot {b}^{2}}\right)\right) \]
      unpow2 [=>] 40.8	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{b \cdot b} - -1 \cdot {b}^{2}}\right)\right) \]
      mul-1-neg [=>] 40.8	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{b \cdot b - \color{blue}{\left(-{b}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 40.8	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{b \cdot b - \left(-\color{blue}{b \cdot b}\right)}\right)\right) \]

   7. Applied egg-rr 40.8

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

   8. Applied egg-rr 31.3

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

   9. Simplified 31.3

      \[\leadsto 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \color{blue}{0}\right)\right) \]
      Proof

      [Start] 31.3	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(b - b\right)\right)\right) \]
      +-inverses [=>] 31.3	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \color{blue}{0}\right)\right) \]

   if 1.0000000000000001e-184 < y-scale < 6.19999999999999986e32

   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 62.9

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(\frac{-0.25}{b \cdot a} \cdot \frac{\sqrt{b \cdot \left(\left(a \cdot \left(a \cdot \left(\frac{-8 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(-b\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{x-scale \cdot x-scale} - \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{b \cdot b - a \cdot a}{\frac{x-scale}{2 \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)}} \cdot \frac{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}{y-scale}\right)\right)\right)\right)}}{b \cdot \left(-a\right)}\right)} \]
      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 x-scale around inf 60.0

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - 0.5 \cdot \frac{-2 \cdot \left(\left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) \cdot \left(\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right) + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\left({b}^{2} - {a}^{2}\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}{\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right)} \]

   4. Simplified 60.1

      \[\leadsto \color{blue}{0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right) - 0.5 \cdot \frac{\mathsf{fma}\left(-2, \left(\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right) \cdot \mathsf{fma}\left(b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right), \frac{4 \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\left(b \cdot b - a \cdot a\right)}^{2}}}\right)}{\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}}}\right)\right)} \]
      Proof

      [Start] 60.0

      \[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - 0.5 \cdot \frac{-2 \cdot \left(\left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) \cdot \left(\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right) + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\left({b}^{2} - {a}^{2}\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}{\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right) \]

   5. Taylor expanded in angle around 0 42.7

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

   6. Simplified 42.7

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

      [Start] 42.7	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{{b}^{2} - -1 \cdot {b}^{2}}\right)\right) \]
      unpow2 [=>] 42.7	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{b \cdot b} - -1 \cdot {b}^{2}}\right)\right) \]
      mul-1-neg [=>] 42.7	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{b \cdot b - \color{blue}{\left(-{b}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 42.7	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{b \cdot b - \left(-\color{blue}{b \cdot b}\right)}\right)\right) \]

   7. Applied egg-rr 42.7

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

   8. Applied egg-rr 42.7

      \[\leadsto 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \color{blue}{{\left(b \cdot \left(b + b\right)\right)}^{0.5}}\right)\right) \]

   9. Simplified 39.7

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

      [Start] 42.7	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot {\left(b \cdot \left(b + b\right)\right)}^{0.5}\right)\right) \]
      unpow1/2 [=>] 42.7	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \color{blue}{\sqrt{b \cdot \left(b + b\right)}}\right)\right) \]
      distribute-lft-in [=>] 42.7	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{b \cdot b + b \cdot b}}\right)\right) \]
      hypot-def [=>] 39.7	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \color{blue}{\mathsf{hypot}\left(b, b\right)}\right)\right) \]

   if 7.00000000000000028e122 < y-scale < 3.3e134

   1. Initial program 64.0

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

   2. Simplified 63.6

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

      [Start] 64.0

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

   3. Taylor expanded in x-scale around inf 62.7

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - 0.5 \cdot \frac{-2 \cdot \left(\left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) \cdot \left(\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right) + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\left({b}^{2} - {a}^{2}\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}{\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right)} \]

   4. Simplified 62.7

      \[\leadsto \color{blue}{0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\mathsf{fma}\left(b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right) - 0.5 \cdot \frac{\mathsf{fma}\left(-2, \left(\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}\right) \cdot \mathsf{fma}\left(b \cdot b, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right), \frac{4 \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\left(b \cdot b - a \cdot a\right)}^{2}}}\right)}{\frac{b \cdot b}{\frac{y-scale \cdot y-scale}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}} + \frac{a \cdot a}{\frac{y-scale \cdot y-scale}{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}}}\right)\right)} \]
      Proof

      [Start] 62.7

      \[ 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) - 0.5 \cdot \frac{-2 \cdot \left(\left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) \cdot \left(\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right) + 4 \cdot \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\left({b}^{2} - {a}^{2}\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}{\frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}}\right) \]

   5. Taylor expanded in angle around 0 53.5

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

   6. Simplified 53.5

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

      [Start] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{{b}^{2} - -1 \cdot {b}^{2}}\right)\right) \]
      unpow2 [=>] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{b \cdot b} - -1 \cdot {b}^{2}}\right)\right) \]
      mul-1-neg [=>] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{b \cdot b - \color{blue}{\left(-{b}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{b \cdot b - \left(-\color{blue}{b \cdot b}\right)}\right)\right) \]

   7. Applied egg-rr 53.5

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

   8. Applied egg-rr 53.5

      \[\leadsto 0.25 \cdot \left(y-scale \cdot \color{blue}{{\left(\left(8 \cdot b\right) \cdot \left(b + b\right)\right)}^{0.5}}\right) \]

   9. Simplified 53.5

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

      [Start] 53.5	\[ 0.25 \cdot \left(y-scale \cdot {\left(\left(8 \cdot b\right) \cdot \left(b + b\right)\right)}^{0.5}\right) \]
      unpow1/2 [=>] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \color{blue}{\sqrt{\left(8 \cdot b\right) \cdot \left(b + b\right)}}\right) \]
      associate-*l* [=>] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \sqrt{\color{blue}{8 \cdot \left(b \cdot \left(b + b\right)\right)}}\right) \]
      distribute-lft-in [=>] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \sqrt{8 \cdot \color{blue}{\left(b \cdot b + b \cdot b\right)}}\right) \]
      distribute-rgt-in [=>] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \sqrt{\color{blue}{\left(b \cdot b\right) \cdot 8 + \left(b \cdot b\right) \cdot 8}}\right) \]
      distribute-lft-out [=>] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \sqrt{\color{blue}{\left(b \cdot b\right) \cdot \left(8 + 8\right)}}\right) \]
      metadata-eval [=>] 53.5	\[ 0.25 \cdot \left(y-scale \cdot \sqrt{\left(b \cdot b\right) \cdot \color{blue}{16}}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 40.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -1.5 \cdot 10^{+117}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \mathbf{elif}\;y-scale \leq -1.25 \cdot 10^{-51}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;y-scale \leq 10^{-184}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot 0\right)\right)\\ \mathbf{elif}\;y-scale \leq 6.2 \cdot 10^{+32}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \mathsf{hypot}\left(b, b\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 7 \cdot 10^{+122}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \mathbf{elif}\;y-scale \leq 3.3 \cdot 10^{+134}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]

Alternatives

Alternative 1
Error	41.1
Cost	20172

\[\begin{array}{l} t_0 := y-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot b\right)\right)\\ \mathbf{if}\;x-scale \leq -1.64 \cdot 10^{+41}:\\ \;\;\;\;0.25 \cdot t_0\\ \mathbf{elif}\;x-scale \leq 4.2 \cdot 10^{-140}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot 0\right)\right)\\ \mathbf{elif}\;x-scale \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \left|\sqrt{8} \cdot a\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot -0.25\\ \end{array} \]

Alternative 2
Error	42.1
Cost	7504

\[\begin{array}{l} t_0 := x-scale \cdot \left(-a\right)\\ \mathbf{if}\;y-scale \leq -3.8 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq -9 \cdot 10^{-53}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;y-scale \leq 1.3 \cdot 10^{-184}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot 0\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.35 \cdot 10^{-130}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{\left(b \cdot b\right) \cdot 16}\right)\\ \mathbf{elif}\;y-scale \leq 5.4 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]

Alternative 3
Error	41.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-11}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-53}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]

Alternative 4
Error	49.7
Cost	653

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -1.7 \cdot 10^{+117} \lor \neg \left(y-scale \leq -5.2 \cdot 10^{-139}\right) \land y-scale \leq 1.55 \cdot 10^{+165}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]

Alternative 5
Error	49.7
Cost	192

\[x-scale \cdot a \]

Reproduce

herbie shell --seed 2023016 
(FPCore (a b angle x-scale y-scale)
  :name "b 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))))