b from scale-rotated-ellipse

Average Accuracy: 0.1% → 41.7%

Time: 2.3min

Precision: binary64

Cost: 84872

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}^{2}\\ t_1 := a \cdot \sqrt{2}\\ \mathbf{if}\;y-scale \leq -3 \cdot 10^{-39}:\\ \;\;\;\;\left|\left(\sqrt{8} \cdot t_1\right) \cdot \left(0.25 \cdot x-scale\right)\right|\\ \mathbf{elif}\;y-scale \leq 3.9 \cdot 10^{-114}:\\ \;\;\;\;-0.25 \cdot \log \left({\left({\left({\left(e^{y-scale}\right)}^{\left(\sqrt{8} \cdot a\right)}\right)}^{x-scale}\right)}^{\left(\sqrt{t_0 + t_0}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\left(0.25 \cdot x-scale\right) \cdot \left({8}^{0.25} \cdot \left(t_1 \cdot {8}^{0.25}\right)\right)\right|\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (*
      (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0)))
      (* (* b a) (* b (- a))))
     (-
      (+
       (/
        (/
         (+
          (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
          (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
         x-scale)
        x-scale)
       (/
        (/
         (+
          (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
          (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
         y-scale)
        y-scale))
      (sqrt
       (+
        (pow
         (-
          (/
           (/
            (+
             (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
             (pow (* b (cos (* (/ angle 180.0) PI))) 2.0))
            x-scale)
           x-scale)
          (/
           (/
            (+
             (pow (* a (cos (* (/ angle 180.0) PI))) 2.0)
             (pow (* b (sin (* (/ angle 180.0) PI))) 2.0))
            y-scale)
           y-scale))
         2.0)
        (pow
         (/
          (/
           (*
            (*
             (* 2.0 (- (pow b 2.0) (pow a 2.0)))
             (sin (* (/ angle 180.0) PI)))
            (cos (* (/ angle 180.0) PI)))
           x-scale)
          y-scale)
         2.0)))))))
  (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (/ (sin (* 0.005555555555555556 (* angle PI))) x-scale) 2.0))
        (t_1 (* a (sqrt 2.0))))
   (if (<= y-scale -3e-39)
     (fabs (* (* (sqrt 8.0) t_1) (* 0.25 x-scale)))
     (if (<= y-scale 3.9e-114)
       (*
        -0.25
        (log
         (pow
          (pow (pow (exp y-scale) (* (sqrt 8.0) a)) x-scale)
          (sqrt (+ t_0 t_0)))))
       (fabs
        (* (* 0.25 x-scale) (* (pow 8.0 0.25) (* t_1 (pow 8.0 0.25)))))))))

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 = pow((sin((0.005555555555555556 * (angle * ((double) M_PI)))) / x_45_scale), 2.0);
	double t_1 = a * sqrt(2.0);
	double tmp;
	if (y_45_scale <= -3e-39) {
		tmp = fabs(((sqrt(8.0) * t_1) * (0.25 * x_45_scale)));
	} else if (y_45_scale <= 3.9e-114) {
		tmp = -0.25 * log(pow(pow(pow(exp(y_45_scale), (sqrt(8.0) * a)), x_45_scale), sqrt((t_0 + t_0))));
	} else {
		tmp = fabs(((0.25 * x_45_scale) * (pow(8.0, 0.25) * (t_1 * pow(8.0, 0.25)))));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow((Math.sin((0.005555555555555556 * (angle * Math.PI))) / x_45_scale), 2.0);
	double t_1 = a * Math.sqrt(2.0);
	double tmp;
	if (y_45_scale <= -3e-39) {
		tmp = Math.abs(((Math.sqrt(8.0) * t_1) * (0.25 * x_45_scale)));
	} else if (y_45_scale <= 3.9e-114) {
		tmp = -0.25 * Math.log(Math.pow(Math.pow(Math.pow(Math.exp(y_45_scale), (Math.sqrt(8.0) * a)), x_45_scale), Math.sqrt((t_0 + t_0))));
	} else {
		tmp = Math.abs(((0.25 * x_45_scale) * (Math.pow(8.0, 0.25) * (t_1 * Math.pow(8.0, 0.25)))));
	}
	return tmp;
}

Python

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

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow((math.sin((0.005555555555555556 * (angle * math.pi))) / x_45_scale), 2.0)
	t_1 = a * math.sqrt(2.0)
	tmp = 0
	if y_45_scale <= -3e-39:
		tmp = math.fabs(((math.sqrt(8.0) * t_1) * (0.25 * x_45_scale)))
	elif y_45_scale <= 3.9e-114:
		tmp = -0.25 * math.log(math.pow(math.pow(math.pow(math.exp(y_45_scale), (math.sqrt(8.0) * a)), x_45_scale), math.sqrt((t_0 + t_0))))
	else:
		tmp = math.fabs(((0.25 * x_45_scale) * (math.pow(8.0, 0.25) * (t_1 * math.pow(8.0, 0.25)))))
	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(sin(Float64(0.005555555555555556 * Float64(angle * pi))) / x_45_scale) ^ 2.0
	t_1 = Float64(a * sqrt(2.0))
	tmp = 0.0
	if (y_45_scale <= -3e-39)
		tmp = abs(Float64(Float64(sqrt(8.0) * t_1) * Float64(0.25 * x_45_scale)));
	elseif (y_45_scale <= 3.9e-114)
		tmp = Float64(-0.25 * log((((exp(y_45_scale) ^ Float64(sqrt(8.0) * a)) ^ x_45_scale) ^ sqrt(Float64(t_0 + t_0)))));
	else
		tmp = abs(Float64(Float64(0.25 * x_45_scale) * Float64((8.0 ^ 0.25) * Float64(t_1 * (8.0 ^ 0.25)))));
	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 = (sin((0.005555555555555556 * (angle * pi))) / x_45_scale) ^ 2.0;
	t_1 = a * sqrt(2.0);
	tmp = 0.0;
	if (y_45_scale <= -3e-39)
		tmp = abs(((sqrt(8.0) * t_1) * (0.25 * x_45_scale)));
	elseif (y_45_scale <= 3.9e-114)
		tmp = -0.25 * log((((exp(y_45_scale) ^ (sqrt(8.0) * a)) ^ x_45_scale) ^ sqrt((t_0 + t_0))));
	else
		tmp = abs(((0.25 * x_45_scale) * ((8.0 ^ 0.25) * (t_1 * (8.0 ^ 0.25)))));
	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[Power[N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -3e-39], N[Abs[N[(N[(N[Sqrt[8.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.25 * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$45$scale, 3.9e-114], N[(-0.25 * N[Log[N[Power[N[Power[N[Power[N[Exp[y$45$scale], $MachinePrecision], N[(N[Sqrt[8.0], $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision], x$45$scale], $MachinePrecision], N[Sqrt[N[(t$95$0 + t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(0.25 * x$45$scale), $MachinePrecision] * N[(N[Power[8.0, 0.25], $MachinePrecision] * N[(t$95$1 * N[Power[8.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < -3.00000000000000028e-39

   1. Initial program 0.1%

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

   2. Simplified 1.4%

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

      [Start] 0.1

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

   3. Taylor expanded in angle around 0 26.1%

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

   4. Simplified 26.1%

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

      [Start] 26.1	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      *-commutative [=>] 26.1	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)}\right) \]
      *-commutative [=>] 26.1	\[ 0.25 \cdot \left(x-scale \cdot \left(\color{blue}{\left(\sqrt{8} \cdot a\right)} \cdot \sqrt{2}\right)\right) \]

   5. Applied egg-rr 37.5%

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

   if -3.00000000000000028e-39 < y-scale < 3.90000000000000002e-114

   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. Simplified 1.0%

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

      [Start] 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 0.9%

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

   4. Simplified 0.9%

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

      [Start] 0.9

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

      *-commutative [=>] 0.9

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

   5. Taylor expanded in y-scale around inf 30.1%

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

   6. Simplified 31.3%

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

      [Start] 30.1	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + -1 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}}\right) \]
      mul-1-neg [=>] 30.1	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + \color{blue}{\left(-\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}}\right) \]
      unpow2 [=>] 30.1	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + \left(-\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right)}\right) \]
      unpow2 [=>] 30.1	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + \left(-\frac{\color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}{x-scale \cdot x-scale}\right)}\right) \]
      unpow2 [<=] 30.1	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + \left(-\frac{\color{blue}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}}{x-scale \cdot x-scale}\right)}\right) \]
      *-commutative [=>] 30.1	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + \left(-\frac{{\sin \left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)}^{2}}{x-scale \cdot x-scale}\right)}\right) \]
      associate-*r* [=>] 31.3	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + \left(-\frac{{\sin \color{blue}{\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)}}^{2}}{x-scale \cdot x-scale}\right)}\right) \]
      *-commutative [<=] 31.3	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + \left(-\frac{{\sin \color{blue}{\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}^{2}}{x-scale \cdot x-scale}\right)}\right) \]
      *-commutative [=>] 31.3	\[ -0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}}{x-scale \cdot x-scale} + \left(-\frac{{\sin \left(angle \cdot \color{blue}{\left(\pi \cdot 0.005555555555555556\right)}\right)}^{2}}{x-scale \cdot x-scale}\right)}\right) \]

   7. Applied egg-rr 50.0%

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

   if 3.90000000000000002e-114 < y-scale

   1. Initial program 0.1%

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

   2. Simplified 1.3%

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

      [Start] 0.1

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

   3. Taylor expanded in angle around 0 25.1%

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

   4. Simplified 25.1%

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

      [Start] 25.1	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      *-commutative [=>] 25.1	\[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)}\right) \]
      *-commutative [=>] 25.1	\[ 0.25 \cdot \left(x-scale \cdot \left(\color{blue}{\left(\sqrt{8} \cdot a\right)} \cdot \sqrt{2}\right)\right) \]

   5. Applied egg-rr 35.8%

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

   6. Applied egg-rr 35.6%

      \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{a \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)}\right)}^{3}} \cdot \left(0.25 \cdot x-scale\right)\right| \]

   7. Applied egg-rr 35.9%

      \[\leadsto \left|\color{blue}{\left(\left(\left(\sqrt{2} \cdot a\right) \cdot {8}^{0.25}\right) \cdot {8}^{0.25}\right)} \cdot \left(0.25 \cdot x-scale\right)\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -3 \cdot 10^{-39}:\\ \;\;\;\;\left|\left(\sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\right) \cdot \left(0.25 \cdot x-scale\right)\right|\\ \mathbf{elif}\;y-scale \leq 3.9 \cdot 10^{-114}:\\ \;\;\;\;-0.25 \cdot \log \left({\left({\left({\left(e^{y-scale}\right)}^{\left(\sqrt{8} \cdot a\right)}\right)}^{x-scale}\right)}^{\left(\sqrt{{\left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}^{2} + {\left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale}\right)}^{2}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\left(0.25 \cdot x-scale\right) \cdot \left({8}^{0.25} \cdot \left(\left(a \cdot \sqrt{2}\right) \cdot {8}^{0.25}\right)\right)\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	41.4%
Cost	26696

\[\begin{array}{l} t_0 := a \cdot \sqrt{2}\\ \mathbf{if}\;y-scale \leq -2.4 \cdot 10^{-39}:\\ \;\;\;\;\left|\left(\sqrt{8} \cdot t_0\right) \cdot \left(0.25 \cdot x-scale\right)\right|\\ \mathbf{elif}\;y-scale \leq 3.8 \cdot 10^{-114}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\left(0.25 \cdot x-scale\right) \cdot \left({8}^{0.25} \cdot \left(t_0 \cdot {8}^{0.25}\right)\right)\right|\\ \end{array} \]

Alternative 2
Accuracy	41.3%
Cost	20041

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -3 \cdot 10^{-39} \lor \neg \left(y-scale \leq 2.7 \cdot 10^{-114}\right):\\ \;\;\;\;\left|\left(\sqrt{8} \cdot \left(a \cdot \sqrt{2}\right)\right) \cdot \left(0.25 \cdot x-scale\right)\right|\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{0}\right)\\ \end{array} \]

Alternative 3
Accuracy	41.2%
Cost	19977

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -2.2 \cdot 10^{-39} \lor \neg \left(y-scale \leq 1.5 \cdot 10^{-114}\right):\\ \;\;\;\;\left|\left(0.25 \cdot x-scale\right) \cdot \left(a \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{0}\right)\\ \end{array} \]

Alternative 4
Accuracy	30.4%
Cost	14037

\[\begin{array}{l} t_0 := \left|\left(0.25 \cdot x-scale\right) \cdot \sqrt{a \cdot \left(a \cdot 16\right)}\right|\\ \mathbf{if}\;b \leq -2.9 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(\left(a \cdot 4 + 1\right) + -1\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+176} \lor \neg \left(b \leq 1.3 \cdot 10^{+229}\right) \land b \leq 1.7 \cdot 10^{+277}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 5
Accuracy	37.9%
Cost	13905

\[\begin{array}{l} t_0 := \left|\left(0.25 \cdot x-scale\right) \cdot \sqrt{a \cdot \left(a \cdot 16\right)}\right|\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-48}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot a\right)\right)\right) \cdot \sqrt{0}\right)\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+175} \lor \neg \left(b \leq 1.06 \cdot 10^{+225}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 6
Accuracy	27.6%
Cost	969

\[\begin{array}{l} \mathbf{if}\;angle \leq -9 \cdot 10^{-205} \lor \neg \left(angle \leq 1.5 \cdot 10^{-119}\right):\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(\left(a \cdot 4 + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(a \cdot -4\right)\\ \end{array} \]

Alternative 7
Accuracy	22.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{+24} \lor \neg \left(b \leq 1.35 \cdot 10^{+175}\right):\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(a \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(a \cdot -4\right)\\ \end{array} \]

Alternative 8
Accuracy	22.9%
Cost	448

\[\left(0.25 \cdot x-scale\right) \cdot \left(a \cdot -4\right) \]

Reproduce

herbie shell --seed 2023129 
(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))))