a from scale-rotated-ellipse

Average Error: 99.22% → 64.27%

Time: 1.4min

Precision: binary64

Cost: 59473

Math

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

↓

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := a \cdot \cos t_0\\ \mathbf{if}\;x-scale \leq -1.8 \cdot 10^{+26}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 8 \cdot 10^{-131} \lor \neg \left(x-scale \leq 3.3 \cdot 10^{-44}\right) \land x-scale \leq 4.8 \cdot 10^{-26}:\\ \;\;\;\;\left|y-scale \cdot b\right|\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot {\left(\sqrt{\sqrt{2} \cdot \mathsf{hypot}\left(b \cdot \sin t_0, t_1\right)}\right)}^{2}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI)) (t_1 (* a (cos t_0))))
   (if (<= x-scale -1.8e+26)
     (* -0.25 (* x-scale (* (sqrt 8.0) (* (sqrt 2.0) t_1))))
     (if (or (<= x-scale 8e-131)
             (and (not (<= x-scale 3.3e-44)) (<= x-scale 4.8e-26)))
       (fabs (* y-scale b))
       (*
        0.25
        (*
         x-scale
         (*
          (sqrt 8.0)
          (pow (sqrt (* (sqrt 2.0) (hypot (* b (sin t_0)) t_1))) 2.0))))))))

C

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

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = a * cos(t_0);
	double tmp;
	if (x_45_scale <= -1.8e+26) {
		tmp = -0.25 * (x_45_scale * (sqrt(8.0) * (sqrt(2.0) * t_1)));
	} else if ((x_45_scale <= 8e-131) || (!(x_45_scale <= 3.3e-44) && (x_45_scale <= 4.8e-26))) {
		tmp = fabs((y_45_scale * b));
	} else {
		tmp = 0.25 * (x_45_scale * (sqrt(8.0) * pow(sqrt((sqrt(2.0) * hypot((b * sin(t_0)), t_1))), 2.0)));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (0.005555555555555556 * angle) * Math.PI;
	double t_1 = a * Math.cos(t_0);
	double tmp;
	if (x_45_scale <= -1.8e+26) {
		tmp = -0.25 * (x_45_scale * (Math.sqrt(8.0) * (Math.sqrt(2.0) * t_1)));
	} else if ((x_45_scale <= 8e-131) || (!(x_45_scale <= 3.3e-44) && (x_45_scale <= 4.8e-26))) {
		tmp = Math.abs((y_45_scale * b));
	} else {
		tmp = 0.25 * (x_45_scale * (Math.sqrt(8.0) * Math.pow(Math.sqrt((Math.sqrt(2.0) * Math.hypot((b * Math.sin(t_0)), t_1))), 2.0)));
	}
	return tmp;
}

Python

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

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (0.005555555555555556 * angle) * math.pi
	t_1 = a * math.cos(t_0)
	tmp = 0
	if x_45_scale <= -1.8e+26:
		tmp = -0.25 * (x_45_scale * (math.sqrt(8.0) * (math.sqrt(2.0) * t_1)))
	elif (x_45_scale <= 8e-131) or (not (x_45_scale <= 3.3e-44) and (x_45_scale <= 4.8e-26)):
		tmp = math.fabs((y_45_scale * b))
	else:
		tmp = 0.25 * (x_45_scale * (math.sqrt(8.0) * math.pow(math.sqrt((math.sqrt(2.0) * math.hypot((b * math.sin(t_0)), t_1))), 2.0)))
	return tmp

Julia

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

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * pi)
	t_1 = Float64(a * cos(t_0))
	tmp = 0.0
	if (x_45_scale <= -1.8e+26)
		tmp = Float64(-0.25 * Float64(x_45_scale * Float64(sqrt(8.0) * Float64(sqrt(2.0) * t_1))));
	elseif ((x_45_scale <= 8e-131) || (!(x_45_scale <= 3.3e-44) && (x_45_scale <= 4.8e-26)))
		tmp = abs(Float64(y_45_scale * b));
	else
		tmp = Float64(0.25 * Float64(x_45_scale * Float64(sqrt(8.0) * (sqrt(Float64(sqrt(2.0) * hypot(Float64(b * sin(t_0)), t_1))) ^ 2.0))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (0.005555555555555556 * angle) * pi;
	t_1 = a * cos(t_0);
	tmp = 0.0;
	if (x_45_scale <= -1.8e+26)
		tmp = -0.25 * (x_45_scale * (sqrt(8.0) * (sqrt(2.0) * t_1)));
	elseif ((x_45_scale <= 8e-131) || (~((x_45_scale <= 3.3e-44)) && (x_45_scale <= 4.8e-26)))
		tmp = abs((y_45_scale * b));
	else
		tmp = 0.25 * (x_45_scale * (sqrt(8.0) * (sqrt((sqrt(2.0) * hypot((b * sin(t_0)), t_1))) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.8e+26], N[(-0.25 * N[(x$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$45$scale, 8e-131], And[N[Not[LessEqual[x$45$scale, 3.3e-44]], $MachinePrecision], LessEqual[x$45$scale, 4.8e-26]]], N[Abs[N[(y$45$scale * b), $MachinePrecision]], $MachinePrecision], N[(0.25 * N[(x$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -1.80000000000000012e26

   1. Initial program 98.74

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

   2. Taylor expanded in y-scale around 0 98.97

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

   3. Simplified 98.97

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

      [Start] 98.97	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      associate-*l* [=>] 98.97	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
      distribute-lft-out [=>] 98.97	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      +-commutative [=>] 98.97	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      fma-def [=>] 98.97	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({b}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 98.97	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right) \]

   4. Taylor expanded in a around -inf 81.58

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

   5. Simplified 81.64

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

      [Start] 81.58	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\right) \]
      mul-1-neg [=>] 81.58	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \color{blue}{\left(-\sqrt{2} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)\right) \]
      *-commutative [=>] 81.58	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(-\sqrt{2} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right)\right)\right)\right)\right) \]
      associate-*l* [<=] 81.58	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(-\sqrt{2} \cdot \left(a \cdot \cos \color{blue}{\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)}\right)\right)\right)\right) \]
      associate-*l* [=>] 81.58	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(-\sqrt{2} \cdot \left(a \cdot \cos \color{blue}{\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right)\right)\right)\right) \]
      *-commutative [<=] 81.58	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(-\sqrt{2} \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)\right)\right)\right) \]
      associate-*r* [=>] 81.64	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(-\sqrt{2} \cdot \left(a \cdot \cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right)\right)\right)\right) \]

   if -1.80000000000000012e26 < x-scale < 7.9999999999999999e-131 or 3.30000000000000006e-44 < x-scale < 4.8000000000000002e-26

   1. Initial program 99.37

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

   2. Simplified 99.17

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

      [Start] 99.37

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

   3. Taylor expanded in angle around 0 79.31

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

   4. Simplified 79.31

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

      [Start] 79.31	\[ 0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 79.31	\[ \color{blue}{\left(0.25 \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \left(b \cdot \sqrt{8}\right)\right)} \]
      associate-*r* [=>] 79.31	\[ \left(0.25 \cdot y-scale\right) \cdot \color{blue}{\left(\left(\sqrt{2} \cdot b\right) \cdot \sqrt{8}\right)} \]
      *-commutative [=>] 79.31	\[ \left(0.25 \cdot y-scale\right) \cdot \left(\color{blue}{\left(b \cdot \sqrt{2}\right)} \cdot \sqrt{8}\right) \]

   5. Applied egg-rr 62.75

      \[\leadsto \color{blue}{\left|\left(b \cdot 4\right) \cdot \left(0.25 \cdot y-scale\right)\right|} \]

   6. Taylor expanded in b around 0 62.75

      \[\leadsto \left|\color{blue}{y-scale \cdot b}\right| \]

   if 7.9999999999999999e-131 < x-scale < 3.30000000000000006e-44 or 4.8000000000000002e-26 < x-scale

   1. Initial program 99.25

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

   2. Taylor expanded in y-scale around 0 74.07

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

   3. Simplified 74.03

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

      [Start] 74.07	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      associate-*l* [=>] 74.03	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
      distribute-lft-out [=>] 74.03	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      +-commutative [=>] 74.03	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      fma-def [=>] 74.03	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({b}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 74.03	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right) \]

   4. Applied egg-rr 84.18

      \[\leadsto 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2}\right)} - \left(1 - {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}^{2}\right)\right)}}\right)\right) \]

   5. Simplified 70.93

      \[\leadsto 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\left({\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\right)\right)\right)}}\right)\right) \]
      Proof

      [Start] 84.18	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left({\left(\cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2}\right)} - \left(1 - {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}^{2}\right)\right)}\right)\right) \]
      associate--r- [=>] 80.25	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(e^{\mathsf{log1p}\left({\left(\cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2}\right)} - 1\right) + {\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}^{2}\right)}}\right)\right) \]
      +-commutative [=>] 80.25	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\left({\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}^{2} + \left(e^{\mathsf{log1p}\left({\left(\cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2}\right)} - 1\right)\right)}}\right)\right) \]
      *-commutative [=>] 80.25	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(b \cdot \sin \left(\color{blue}{\left(\pi \cdot 0.005555555555555556\right)} \cdot angle\right)\right)}^{2} + \left(e^{\mathsf{log1p}\left({\left(\cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2}\right)} - 1\right)\right)}\right)\right) \]
      associate-*l* [=>] 80.27	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(b \cdot \sin \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2} + \left(e^{\mathsf{log1p}\left({\left(\cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2}\right)} - 1\right)\right)}\right)\right) \]
      *-commutative [=>] 80.27	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(b \cdot \sin \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2} + \left(e^{\mathsf{log1p}\left({\left(\cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2}\right)} - 1\right)\right)}\right)\right) \]
      expm1-def [=>] 70.93	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right) \cdot a\right)}^{2}\right)\right)}\right)}\right)\right) \]
      *-commutative [=>] 70.93	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \mathsf{expm1}\left(\mathsf{log1p}\left({\color{blue}{\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}}^{2}\right)\right)\right)}\right)\right) \]
      *-commutative [=>] 70.93	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot \cos \left(\color{blue}{\left(\pi \cdot 0.005555555555555556\right)} \cdot angle\right)\right)}^{2}\right)\right)\right)}\right)\right) \]
      associate-*l* [=>] 70.93	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot \cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)}^{2}\right)\right)\right)}\right)\right) \]
      *-commutative [=>] 70.93	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)}^{2}\right)\right)\right)}\right)\right) \]

   6. Applied egg-rr 56.92

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

4. Final simplification 64.27

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.8 \cdot 10^{+26}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 8 \cdot 10^{-131} \lor \neg \left(x-scale \leq 3.3 \cdot 10^{-44}\right) \land x-scale \leq 4.8 \cdot 10^{-26}:\\ \;\;\;\;\left|y-scale \cdot b\right|\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot {\left(\sqrt{\sqrt{2} \cdot \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}\right)}^{2}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	64.33%
Cost	46609

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := a \cdot \cos t_0\\ \mathbf{if}\;x-scale \leq -5.1 \cdot 10^{+24}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 8.2 \cdot 10^{-127} \lor \neg \left(x-scale \leq 1.25 \cdot 10^{-42}\right) \land x-scale \leq 10^{-25}:\\ \;\;\;\;\left|y-scale \cdot b\right|\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(t_1, b \cdot \sin t_0\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	70.87%
Cost	46412

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := \cos t_0\\ \mathbf{if}\;y-scale \leq -1.05 \cdot 10^{-114}:\\ \;\;\;\;\left|y-scale \cdot b\right|\\ \mathbf{elif}\;y-scale \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(a \cdot \left(\sqrt{2} \cdot t_1\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.75 \cdot 10^{-42}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right) \cdot \sqrt{\frac{{t_1}^{2}}{y-scale \cdot y-scale} + \frac{\frac{1}{y-scale}}{y-scale}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot y-scale\right) \cdot \mathsf{hypot}\left(\mathsf{hypot}\left(b, a \cdot \sin t_0\right), b \cdot t_1\right)\right)\\ \end{array} \]

Alternative 3
Error	72.47%
Cost	27552

\[\begin{array}{l} t_0 := \left|y-scale \cdot b\right|\\ t_1 := \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\\ t_2 := -0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \left(a \cdot t_1\right)\right)\right)\right)\\ t_3 := 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(a \cdot \left(\sqrt{2} \cdot t_1\right)\right)\right)\right)\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-299}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-309}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-124}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(a \cdot a\right)}\right)\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 28000000:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	71.54%
Cost	26760

\[\begin{array}{l} t_0 := \left|y-scale \cdot b\right|\\ \mathbf{if}\;y-scale \leq -4.6 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 4.1 \cdot 10^{-142}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(a \cdot \left(\sqrt{2} \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.3 \cdot 10^{-42}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	71.46%
Cost	14028

\[\begin{array}{l} t_0 := \left|y-scale \cdot b\right|\\ \mathbf{if}\;y-scale \leq -3.4 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 2.15 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.55 \cdot 10^{-42}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\left(x-scale \cdot \left(\sqrt{8} \cdot a\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	69.95%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -6.2 \cdot 10^{+62} \lor \neg \left(x-scale \leq 1.5 \cdot 10^{+56}\right):\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y-scale \cdot b\right|\\ \end{array} \]

Alternative 7
Error	71.64%
Cost	6592

\[\left|y-scale \cdot b\right| \]

Alternative 8
Error	84.16%
Cost	192

\[y-scale \cdot b \]

Reproduce

herbie shell --seed 2023089 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))