b from scale-rotated-ellipse

Average Accuracy: 0.1% → 32.6%

Time: 2.4min

Precision: binary64

Cost: 73556

Math

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

↓

\[\begin{array}{l} t_0 := \left(x-scale \cdot 0.25\right) \cdot 0\\ t_1 := \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_2 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;x-scale \leq -2.1 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -1.1 \cdot 10^{-118}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot \left(x-scale \cdot y-scale\right)\right) \cdot \mathsf{hypot}\left(t_1 \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right)\right|\\ \mathbf{elif}\;x-scale \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq 5.2 \cdot 10^{-27}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left|a \cdot 4\right|\\ \mathbf{elif}\;x-scale \leq 4 \cdot 10^{-25}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot \sqrt{\frac{{\sin t_2}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\cos t_2}^{2}}{x-scale \cdot x-scale} - \frac{{t_1}^{2}}{x-scale \cdot x-scale}\right)}\right)\\ \mathbf{elif}\;x-scale \leq 9.5 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \sqrt{a \cdot \left(a \cdot 16\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* x-scale 0.25) 0.0))
        (t_1 (cos (* PI (* angle 0.005555555555555556))))
        (t_2 (* 0.005555555555555556 (* PI angle))))
   (if (<= x-scale -2.1e-29)
     t_0
     (if (<= x-scale -1.1e-118)
       (*
        0.25
        (fabs
         (*
          (* (sqrt 8.0) (* x-scale y-scale))
          (hypot (* t_1 (/ a y-scale)) (/ a y-scale)))))
       (if (<= x-scale 7.5e-134)
         t_0
         (if (<= x-scale 5.2e-27)
           (* (* x-scale 0.25) (fabs (* a 4.0)))
           (if (<= x-scale 4e-25)
             (*
              0.25
              (*
               (* x-scale (* y-scale (* (sqrt 8.0) b)))
               (sqrt
                (+
                 (/ (pow (sin t_2) 2.0) (* y-scale y-scale))
                 (-
                  (/ (pow (cos t_2) 2.0) (* x-scale x-scale))
                  (/ (pow t_1 2.0) (* x-scale x-scale)))))))
             (if (<= x-scale 9.5e+107)
               t_0
               (* (* x-scale 0.25) (sqrt (* a (* a 16.0))))))))))))

C

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

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale * 0.25) * 0.0;
	double t_1 = cos((((double) M_PI) * (angle * 0.005555555555555556)));
	double t_2 = 0.005555555555555556 * (((double) M_PI) * angle);
	double tmp;
	if (x_45_scale <= -2.1e-29) {
		tmp = t_0;
	} else if (x_45_scale <= -1.1e-118) {
		tmp = 0.25 * fabs(((sqrt(8.0) * (x_45_scale * y_45_scale)) * hypot((t_1 * (a / y_45_scale)), (a / y_45_scale))));
	} else if (x_45_scale <= 7.5e-134) {
		tmp = t_0;
	} else if (x_45_scale <= 5.2e-27) {
		tmp = (x_45_scale * 0.25) * fabs((a * 4.0));
	} else if (x_45_scale <= 4e-25) {
		tmp = 0.25 * ((x_45_scale * (y_45_scale * (sqrt(8.0) * b))) * sqrt(((pow(sin(t_2), 2.0) / (y_45_scale * y_45_scale)) + ((pow(cos(t_2), 2.0) / (x_45_scale * x_45_scale)) - (pow(t_1, 2.0) / (x_45_scale * x_45_scale))))));
	} else if (x_45_scale <= 9.5e+107) {
		tmp = t_0;
	} else {
		tmp = (x_45_scale * 0.25) * sqrt((a * (a * 16.0)));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale * 0.25) * 0.0;
	double t_1 = Math.cos((Math.PI * (angle * 0.005555555555555556)));
	double t_2 = 0.005555555555555556 * (Math.PI * angle);
	double tmp;
	if (x_45_scale <= -2.1e-29) {
		tmp = t_0;
	} else if (x_45_scale <= -1.1e-118) {
		tmp = 0.25 * Math.abs(((Math.sqrt(8.0) * (x_45_scale * y_45_scale)) * Math.hypot((t_1 * (a / y_45_scale)), (a / y_45_scale))));
	} else if (x_45_scale <= 7.5e-134) {
		tmp = t_0;
	} else if (x_45_scale <= 5.2e-27) {
		tmp = (x_45_scale * 0.25) * Math.abs((a * 4.0));
	} else if (x_45_scale <= 4e-25) {
		tmp = 0.25 * ((x_45_scale * (y_45_scale * (Math.sqrt(8.0) * b))) * Math.sqrt(((Math.pow(Math.sin(t_2), 2.0) / (y_45_scale * y_45_scale)) + ((Math.pow(Math.cos(t_2), 2.0) / (x_45_scale * x_45_scale)) - (Math.pow(t_1, 2.0) / (x_45_scale * x_45_scale))))));
	} else if (x_45_scale <= 9.5e+107) {
		tmp = t_0;
	} else {
		tmp = (x_45_scale * 0.25) * Math.sqrt((a * (a * 16.0)));
	}
	return tmp;
}

Python

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

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (x_45_scale * 0.25) * 0.0
	t_1 = math.cos((math.pi * (angle * 0.005555555555555556)))
	t_2 = 0.005555555555555556 * (math.pi * angle)
	tmp = 0
	if x_45_scale <= -2.1e-29:
		tmp = t_0
	elif x_45_scale <= -1.1e-118:
		tmp = 0.25 * math.fabs(((math.sqrt(8.0) * (x_45_scale * y_45_scale)) * math.hypot((t_1 * (a / y_45_scale)), (a / y_45_scale))))
	elif x_45_scale <= 7.5e-134:
		tmp = t_0
	elif x_45_scale <= 5.2e-27:
		tmp = (x_45_scale * 0.25) * math.fabs((a * 4.0))
	elif x_45_scale <= 4e-25:
		tmp = 0.25 * ((x_45_scale * (y_45_scale * (math.sqrt(8.0) * b))) * math.sqrt(((math.pow(math.sin(t_2), 2.0) / (y_45_scale * y_45_scale)) + ((math.pow(math.cos(t_2), 2.0) / (x_45_scale * x_45_scale)) - (math.pow(t_1, 2.0) / (x_45_scale * x_45_scale))))))
	elif x_45_scale <= 9.5e+107:
		tmp = t_0
	else:
		tmp = (x_45_scale * 0.25) * math.sqrt((a * (a * 16.0)))
	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(x_45_scale * 0.25) * 0.0)
	t_1 = cos(Float64(pi * Float64(angle * 0.005555555555555556)))
	t_2 = Float64(0.005555555555555556 * Float64(pi * angle))
	tmp = 0.0
	if (x_45_scale <= -2.1e-29)
		tmp = t_0;
	elseif (x_45_scale <= -1.1e-118)
		tmp = Float64(0.25 * abs(Float64(Float64(sqrt(8.0) * Float64(x_45_scale * y_45_scale)) * hypot(Float64(t_1 * Float64(a / y_45_scale)), Float64(a / y_45_scale)))));
	elseif (x_45_scale <= 7.5e-134)
		tmp = t_0;
	elseif (x_45_scale <= 5.2e-27)
		tmp = Float64(Float64(x_45_scale * 0.25) * abs(Float64(a * 4.0)));
	elseif (x_45_scale <= 4e-25)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale * Float64(y_45_scale * Float64(sqrt(8.0) * b))) * sqrt(Float64(Float64((sin(t_2) ^ 2.0) / Float64(y_45_scale * y_45_scale)) + Float64(Float64((cos(t_2) ^ 2.0) / Float64(x_45_scale * x_45_scale)) - Float64((t_1 ^ 2.0) / Float64(x_45_scale * x_45_scale)))))));
	elseif (x_45_scale <= 9.5e+107)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_45_scale * 0.25) * sqrt(Float64(a * Float64(a * 16.0))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (x_45_scale * 0.25) * 0.0;
	t_1 = cos((pi * (angle * 0.005555555555555556)));
	t_2 = 0.005555555555555556 * (pi * angle);
	tmp = 0.0;
	if (x_45_scale <= -2.1e-29)
		tmp = t_0;
	elseif (x_45_scale <= -1.1e-118)
		tmp = 0.25 * abs(((sqrt(8.0) * (x_45_scale * y_45_scale)) * hypot((t_1 * (a / y_45_scale)), (a / y_45_scale))));
	elseif (x_45_scale <= 7.5e-134)
		tmp = t_0;
	elseif (x_45_scale <= 5.2e-27)
		tmp = (x_45_scale * 0.25) * abs((a * 4.0));
	elseif (x_45_scale <= 4e-25)
		tmp = 0.25 * ((x_45_scale * (y_45_scale * (sqrt(8.0) * b))) * sqrt((((sin(t_2) ^ 2.0) / (y_45_scale * y_45_scale)) + (((cos(t_2) ^ 2.0) / (x_45_scale * x_45_scale)) - ((t_1 ^ 2.0) / (x_45_scale * x_45_scale))))));
	elseif (x_45_scale <= 9.5e+107)
		tmp = t_0;
	else
		tmp = (x_45_scale * 0.25) * sqrt((a * (a * 16.0)));
	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[(x$45$scale * 0.25), $MachinePrecision] * 0.0), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -2.1e-29], t$95$0, If[LessEqual[x$45$scale, -1.1e-118], N[(0.25 * N[Abs[N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a / y$45$scale), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 7.5e-134], t$95$0, If[LessEqual[x$45$scale, 5.2e-27], N[(N[(x$45$scale * 0.25), $MachinePrecision] * N[Abs[N[(a * 4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 4e-25], N[(0.25 * N[(N[(x$45$scale * N[(y$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Cos[t$95$2], $MachinePrecision], 2.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 9.5e+107], t$95$0, N[(N[(x$45$scale * 0.25), $MachinePrecision] * N[Sqrt[N[(a * N[(a * 16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x-scale < -2.09999999999999989e-29 or -1.09999999999999992e-118 < x-scale < 7.50000000000000048e-134 or 4.00000000000000015e-25 < x-scale < 9.50000000000000019e107

   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 1.0%

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

      [Start] 0.1

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

   3. Taylor expanded in angle around 0 25.8%

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

   4. Simplified 25.8%

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

      [Start] 25.8	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 25.8	\[ \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 25.8	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]

   5. Applied egg-rr 26.8%

      \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(a \cdot 4\right)} - 1\right)} \]
      Proof

      [Start] 25.8	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right) \]
      expm1-log1p-u [=>] 23.5	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right)} \]
      expm1-udef [=>] 26.8	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} - 1\right)} \]
      associate-*l* [=>] 26.8	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{a \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)}\right)} - 1\right) \]
      sqrt-unprod [=>] 26.8	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)} - 1\right) \]
      metadata-eval [=>] 26.8	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{16}}\right)} - 1\right) \]
      metadata-eval [=>] 26.8	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot \color{blue}{4}\right)} - 1\right) \]

   6. Taylor expanded in a around 0 36.0%

      \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \left(\color{blue}{1} - 1\right) \]

   if -2.09999999999999989e-29 < x-scale < -1.09999999999999992e-118

   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 2.1%

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(\frac{-0.25}{b \cdot a} \cdot \frac{\sqrt{b \cdot \left(\left(a \cdot \left(a \cdot \left(\frac{-8 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(-b\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{x-scale \cdot x-scale} - \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{b \cdot b - a \cdot a}{\frac{x-scale}{2 \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)}} \cdot \frac{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}{y-scale}\right)\right)\right)\right)}}{b \cdot \left(-a\right)}\right)} \]
      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 4.6%

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

   4. Simplified 4.6%

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

      [Start] 4.6	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(\frac{-0.25}{b \cdot a} \cdot \frac{\sqrt{b \cdot \left(\left(a \cdot \left(a \cdot \left(\frac{-8 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(-b\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{{a}^{2}}{{y-scale}^{2}}\right)\right)}}{b \cdot \left(-a\right)}\right) \]
      unpow2 [=>] 4.6	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(\frac{-0.25}{b \cdot a} \cdot \frac{\sqrt{b \cdot \left(\left(a \cdot \left(a \cdot \left(\frac{-8 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(-b\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{\color{blue}{a \cdot a}}{{y-scale}^{2}}\right)\right)}}{b \cdot \left(-a\right)}\right) \]
      unpow2 [=>] 4.6	\[ {\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(\frac{-0.25}{b \cdot a} \cdot \frac{\sqrt{b \cdot \left(\left(a \cdot \left(a \cdot \left(\frac{-8 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(-b\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale} + \frac{a \cdot a}{\color{blue}{y-scale \cdot y-scale}}\right)\right)}}{b \cdot \left(-a\right)}\right) \]

   5. Taylor expanded in b around 0 16.3%

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

   6. Simplified 19.1%

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

      [Start] 16.3	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2}}{{y-scale}^{2}}}\right) \]
      *-commutative [=>] 16.3	\[ \color{blue}{\left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2}}{{y-scale}^{2}}}\right) \cdot 0.25} \]
      associate-*l* [=>] 16.3	\[ \color{blue}{\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2}}{{y-scale}^{2}}} \cdot 0.25\right)} \]
      associate-*r* [=>] 16.3	\[ \color{blue}{\left(\left(x-scale \cdot y-scale\right) \cdot \sqrt{8}\right)} \cdot \left(\sqrt{\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2}}{{y-scale}^{2}}} \cdot 0.25\right) \]
      *-commutative [<=] 16.3	\[ \left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \sqrt{8}\right) \cdot \left(\sqrt{\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2}}{{y-scale}^{2}}} \cdot 0.25\right) \]
      associate-*r* [<=] 16.3	\[ \color{blue}{\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\sqrt{\frac{{a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{a}^{2}}{{y-scale}^{2}}} \cdot 0.25\right) \]

   7. Applied egg-rr 32.4%

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

      [Start] 19.1	\[ \left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{\frac{a}{y-scale \cdot \frac{y-scale}{a}} \cdot {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2} + \frac{a}{y-scale \cdot \frac{y-scale}{a}}} \cdot 0.25\right) \]
      add-sqr-sqrt [=>] 18.3	\[ \color{blue}{\sqrt{\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{\frac{a}{y-scale \cdot \frac{y-scale}{a}} \cdot {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2} + \frac{a}{y-scale \cdot \frac{y-scale}{a}}} \cdot 0.25\right)} \cdot \sqrt{\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{\frac{a}{y-scale \cdot \frac{y-scale}{a}} \cdot {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2} + \frac{a}{y-scale \cdot \frac{y-scale}{a}}} \cdot 0.25\right)}} \]
      sqrt-unprod [=>] 22.4	\[ \color{blue}{\sqrt{\left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{\frac{a}{y-scale \cdot \frac{y-scale}{a}} \cdot {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2} + \frac{a}{y-scale \cdot \frac{y-scale}{a}}} \cdot 0.25\right)\right) \cdot \left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{\frac{a}{y-scale \cdot \frac{y-scale}{a}} \cdot {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2} + \frac{a}{y-scale \cdot \frac{y-scale}{a}}} \cdot 0.25\right)\right)}} \]
      pow2 [=>] 22.4	\[ \sqrt{\color{blue}{{\left(\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{\frac{a}{y-scale \cdot \frac{y-scale}{a}} \cdot {\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}^{2} + \frac{a}{y-scale \cdot \frac{y-scale}{a}}} \cdot 0.25\right)\right)}^{2}}} \]

   8. Simplified 37.9%

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

      [Start] 32.4	\[ \sqrt{{\left(\mathsf{hypot}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right) \cdot \left(0.25 \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot x-scale\right)\right)\right)\right)}^{2}} \]
      unpow2 [=>] 32.4	\[ \sqrt{\color{blue}{\left(\mathsf{hypot}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right) \cdot \left(0.25 \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot x-scale\right)\right)\right)\right) \cdot \left(\mathsf{hypot}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right) \cdot \left(0.25 \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot x-scale\right)\right)\right)\right)}} \]
      rem-sqrt-square [=>] 37.9	\[ \color{blue}{\left|\mathsf{hypot}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right) \cdot \left(0.25 \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot x-scale\right)\right)\right)\right|} \]
      *-commutative [=>] 37.9	\[ \left|\color{blue}{\left(0.25 \cdot \left(\sqrt{8} \cdot \left(y-scale \cdot x-scale\right)\right)\right) \cdot \mathsf{hypot}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right)}\right| \]
      associate-*l* [=>] 37.9	\[ \left|\color{blue}{0.25 \cdot \left(\left(\sqrt{8} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \mathsf{hypot}\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right)\right)}\right| \]
      *-commutative [=>] 37.9	\[ \left|0.25 \cdot \left(\left(\sqrt{8} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \mathsf{hypot}\left(\cos \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right) \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right)\right)\right| \]

   if 7.50000000000000048e-134 < x-scale < 5.20000000000000034e-27

   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 3.1%

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(\frac{-0.25}{b \cdot a} \cdot \frac{\sqrt{b \cdot \left(\left(a \cdot \left(a \cdot \left(\frac{-8 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(-b\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{x-scale \cdot x-scale} - \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{b \cdot b - a \cdot a}{\frac{x-scale}{2 \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)}} \cdot \frac{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}{y-scale}\right)\right)\right)\right)}}{b \cdot \left(-a\right)}\right)} \]
      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 26.2%

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

   4. Simplified 26.2%

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

      [Start] 26.2	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 26.2	\[ \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 26.2	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]

   5. Applied egg-rr 26.5%

      \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(a \cdot 4\right)} - 1\right)} \]
      Proof

      [Start] 26.2	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right) \]
      expm1-log1p-u [=>] 25.5	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right)} \]
      expm1-udef [=>] 26.5	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} - 1\right)} \]
      associate-*l* [=>] 26.5	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{a \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)}\right)} - 1\right) \]
      sqrt-unprod [=>] 26.5	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)} - 1\right) \]
      metadata-eval [=>] 26.5	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{16}}\right)} - 1\right) \]
      metadata-eval [=>] 26.5	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot \color{blue}{4}\right)} - 1\right) \]

   6. Applied egg-rr 32.4%

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

      [Start] 26.5	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot 4\right)} - 1\right) \]
      expm1-def [=>] 25.5	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot 4\right)\right)} \]
      expm1-log1p-u [<=] 26.3	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(a \cdot 4\right)} \]
      add-sqr-sqrt [=>] 19.7	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(\sqrt{a \cdot 4} \cdot \sqrt{a \cdot 4}\right)} \]
      sqrt-unprod [=>] 32.3	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\sqrt{\left(a \cdot 4\right) \cdot \left(a \cdot 4\right)}} \]
      *-commutative [=>] 32.3	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{\left(4 \cdot a\right)} \cdot \left(a \cdot 4\right)} \]
      *-commutative [=>] 32.3	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\left(4 \cdot a\right) \cdot \color{blue}{\left(4 \cdot a\right)}} \]
      swap-sqr [=>] 32.4	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{\left(4 \cdot 4\right) \cdot \left(a \cdot a\right)}} \]
      metadata-eval [=>] 32.4	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{16} \cdot \left(a \cdot a\right)} \]

   7. Simplified 36.7%

      \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left|4 \cdot a\right|} \]
      Proof

      [Start] 32.4	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{16 \cdot \left(a \cdot a\right)} \]
      *-commutative [=>] 32.4	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{\left(a \cdot a\right) \cdot 16}} \]
      metadata-eval [<=] 32.4	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\left(a \cdot a\right) \cdot \color{blue}{\left(4 \cdot 4\right)}} \]
      swap-sqr [<=] 32.3	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{\left(a \cdot 4\right) \cdot \left(a \cdot 4\right)}} \]
      rem-sqrt-square [=>] 36.7	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left|a \cdot 4\right|} \]
      *-commutative [<=] 36.7	\[ \left(0.25 \cdot x-scale\right) \cdot \left|\color{blue}{4 \cdot a}\right| \]

   if 5.20000000000000034e-27 < x-scale < 4.00000000000000015e-25

   1. Initial program 0.3%

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

   2. Simplified 0.3%

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

      [Start] 0.3

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

   3. Taylor expanded in b around inf 4.4%

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

   4. Simplified 3.9%

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

      [Start] 4.4

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

   5. Taylor expanded in x-scale around 0 4.0%

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

   6. Simplified 3.8%

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

      [Start] 4.0	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)}\right) \]
      unpow2 [=>] 4.0	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{\color{blue}{x-scale \cdot x-scale}}\right)}\right) \]
      associate-*r* [=>] 3.8	\[ 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\cos \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}^{2}}{x-scale \cdot x-scale}\right)}\right) \]

   if 9.50000000000000019e107 < x-scale

   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 0.8%

      \[\leadsto \color{blue}{{\left(x-scale \cdot y-scale\right)}^{2} \cdot \left(\frac{-0.25}{b \cdot a} \cdot \frac{\sqrt{b \cdot \left(\left(a \cdot \left(a \cdot \left(\frac{-8 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}} \cdot \left(-b\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{x-scale \cdot x-scale} - \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{b \cdot b - a \cdot a}{\frac{x-scale}{2 \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)}} \cdot \frac{\cos \left(\frac{angle}{\frac{180}{\pi}}\right)}{y-scale}\right)\right)\right)\right)}}{b \cdot \left(-a\right)}\right)} \]
      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 angle around 0 7.3%

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

   4. Simplified 7.3%

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

      [Start] 7.3	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      associate-*r* [=>] 7.3	\[ \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \left(a \cdot \sqrt{8}\right)\right)} \]
      *-commutative [=>] 7.3	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} \]

   5. Applied egg-rr 13.1%

      \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(a \cdot 4\right)} - 1\right)} \]
      Proof

      [Start] 7.3	\[ \left(0.25 \cdot x-scale\right) \cdot \left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right) \]
      expm1-log1p-u [=>] 7.1	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right)} \]
      expm1-udef [=>] 13.1	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(a \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)} - 1\right)} \]
      associate-*l* [=>] 13.1	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{a \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)}\right)} - 1\right) \]
      sqrt-unprod [=>] 13.1	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)} - 1\right) \]
      metadata-eval [=>] 13.1	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{16}}\right)} - 1\right) \]
      metadata-eval [=>] 13.1	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot \color{blue}{4}\right)} - 1\right) \]

   6. Applied egg-rr 13.6%

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

      [Start] 13.1	\[ \left(0.25 \cdot x-scale\right) \cdot \left(e^{\mathsf{log1p}\left(a \cdot 4\right)} - 1\right) \]
      expm1-def [=>] 7.2	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot 4\right)\right)} \]
      expm1-log1p-u [<=] 7.3	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(a \cdot 4\right)} \]
      add-sqr-sqrt [=>] 6.7	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\left(\sqrt{a \cdot 4} \cdot \sqrt{a \cdot 4}\right)} \]
      sqrt-unprod [=>] 13.6	\[ \left(0.25 \cdot x-scale\right) \cdot \color{blue}{\sqrt{\left(a \cdot 4\right) \cdot \left(a \cdot 4\right)}} \]
      *-commutative [=>] 13.6	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{\left(4 \cdot a\right)} \cdot \left(a \cdot 4\right)} \]
      *-commutative [=>] 13.6	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\left(4 \cdot a\right) \cdot \color{blue}{\left(4 \cdot a\right)}} \]
      swap-sqr [=>] 13.6	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{\left(4 \cdot 4\right) \cdot \left(a \cdot a\right)}} \]
      metadata-eval [=>] 13.6	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{16} \cdot \left(a \cdot a\right)} \]

   7. Simplified 13.6%

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

      [Start] 13.6	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{16 \cdot \left(a \cdot a\right)} \]
      *-commutative [=>] 13.6	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{\left(a \cdot a\right) \cdot 16}} \]
      associate-*l* [=>] 13.6	\[ \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{a \cdot \left(a \cdot 16\right)}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -2.1 \cdot 10^{-29}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot 0\\ \mathbf{elif}\;x-scale \leq -1.1 \cdot 10^{-118}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot \left(x-scale \cdot y-scale\right)\right) \cdot \mathsf{hypot}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right)\right|\\ \mathbf{elif}\;x-scale \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot 0\\ \mathbf{elif}\;x-scale \leq 5.2 \cdot 10^{-27}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left|a \cdot 4\right|\\ \mathbf{elif}\;x-scale \leq 4 \cdot 10^{-25}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot b\right)\right)\right) \cdot \sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}}{x-scale \cdot x-scale}\right)}\right)\\ \mathbf{elif}\;x-scale \leq 9.5 \cdot 10^{+107}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \sqrt{a \cdot \left(a \cdot 16\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	35.3%
Cost	33480

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot 0\\ \mathbf{elif}\;y-scale \leq 2 \cdot 10^{+113}:\\ \;\;\;\;0.25 \cdot \left|\left(y-scale \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(\frac{a}{y-scale} \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), \frac{a}{y-scale}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 2
Accuracy	35.3%
Cost	33480

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 7 \cdot 10^{-58}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot 0\\ \mathbf{elif}\;y-scale \leq 3.6 \cdot 10^{+113}:\\ \;\;\;\;0.25 \cdot \left|\left(\sqrt{8} \cdot \left(x-scale \cdot y-scale\right)\right) \cdot \mathsf{hypot}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \frac{a}{y-scale}, \frac{a}{y-scale}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 3
Accuracy	33.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;angle \leq -3 \cdot 10^{-226} \lor \neg \left(angle \leq 2.1 \cdot 10^{-34}\right):\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left|a \cdot 4\right|\\ \end{array} \]

Alternative 4
Accuracy	34.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 5
Accuracy	23.0%
Cost	448

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

Reproduce

herbie shell --seed 2023135 
(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))))