a from scale-rotated-ellipse

Average Error: 63.4 → 39.5

Time: 2.0min

Precision: binary64

Cost: 46608

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := {\left({\left(\sqrt[3]{b \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ t_2 := \sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\\ t_3 := \mathsf{hypot}\left(b \cdot \sin t_0, a \cdot \cos t_0\right)\\ t_4 := 0.25 \cdot \left(t_3 \cdot t_2\right)\\ \mathbf{if}\;x-scale \leq -2.3337806862687206 \cdot 10^{-55}:\\ \;\;\;\;\left(-0.25 \cdot t_3\right) \cdot t_2\\ \mathbf{elif}\;x-scale \leq 2.7985085097999705 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x-scale \leq 3.1 \cdot 10^{+173}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

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 (* PI (* angle 0.005555555555555556)))
        (t_1 (pow (pow (cbrt (* b y-scale)) 2.0) 1.5))
        (t_2 (* (sqrt 2.0) (* x-scale (sqrt 8.0))))
        (t_3 (hypot (* b (sin t_0)) (* a (cos t_0))))
        (t_4 (* 0.25 (* t_3 t_2))))
   (if (<= x-scale -2.3337806862687206e-55)
     (* (* -0.25 t_3) t_2)
     (if (<= x-scale 2.7985085097999705e-87)
       t_1
       (if (<= x-scale 7.5e+48) t_4 (if (<= x-scale 3.1e+173) t_1 t_4))))))

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 = ((double) M_PI) * (angle * 0.005555555555555556);
	double t_1 = pow(pow(cbrt((b * y_45_scale)), 2.0), 1.5);
	double t_2 = sqrt(2.0) * (x_45_scale * sqrt(8.0));
	double t_3 = hypot((b * sin(t_0)), (a * cos(t_0)));
	double t_4 = 0.25 * (t_3 * t_2);
	double tmp;
	if (x_45_scale <= -2.3337806862687206e-55) {
		tmp = (-0.25 * t_3) * t_2;
	} else if (x_45_scale <= 2.7985085097999705e-87) {
		tmp = t_1;
	} else if (x_45_scale <= 7.5e+48) {
		tmp = t_4;
	} else if (x_45_scale <= 3.1e+173) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.PI * (angle * 0.005555555555555556);
	double t_1 = Math.pow(Math.pow(Math.cbrt((b * y_45_scale)), 2.0), 1.5);
	double t_2 = Math.sqrt(2.0) * (x_45_scale * Math.sqrt(8.0));
	double t_3 = Math.hypot((b * Math.sin(t_0)), (a * Math.cos(t_0)));
	double t_4 = 0.25 * (t_3 * t_2);
	double tmp;
	if (x_45_scale <= -2.3337806862687206e-55) {
		tmp = (-0.25 * t_3) * t_2;
	} else if (x_45_scale <= 2.7985085097999705e-87) {
		tmp = t_1;
	} else if (x_45_scale <= 7.5e+48) {
		tmp = t_4;
	} else if (x_45_scale <= 3.1e+173) {
		tmp = t_1;
	} else {
		tmp = t_4;
	}
	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(pi * Float64(angle * 0.005555555555555556))
	t_1 = (cbrt(Float64(b * y_45_scale)) ^ 2.0) ^ 1.5
	t_2 = Float64(sqrt(2.0) * Float64(x_45_scale * sqrt(8.0)))
	t_3 = hypot(Float64(b * sin(t_0)), Float64(a * cos(t_0)))
	t_4 = Float64(0.25 * Float64(t_3 * t_2))
	tmp = 0.0
	if (x_45_scale <= -2.3337806862687206e-55)
		tmp = Float64(Float64(-0.25 * t_3) * t_2);
	elseif (x_45_scale <= 2.7985085097999705e-87)
		tmp = t_1;
	elseif (x_45_scale <= 7.5e+48)
		tmp = t_4;
	elseif (x_45_scale <= 3.1e+173)
		tmp = t_1;
	else
		tmp = t_4;
	end
	return 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[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[Power[N[(b * y$45$scale), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 1.5], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(x$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(0.25 * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -2.3337806862687206e-55], N[(N[(-0.25 * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[x$45$scale, 2.7985085097999705e-87], t$95$1, If[LessEqual[x$45$scale, 7.5e+48], t$95$4, If[LessEqual[x$45$scale, 3.1e+173], t$95$1, t$95$4]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -2.33378068626872063e-55

   1. Initial program 63.2

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

   2. Taylor expanded in x-scale around inf 59.9

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

   3. Simplified 59.0

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

   4. Taylor expanded in y-scale around -inf 45.8

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

   5. Simplified 35.3

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

   if -2.33378068626872063e-55 < x-scale < 2.7985085097999705e-87 or 7.5000000000000006e48 < x-scale < 3.1e173

   1. Initial program 63.6

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

   2. Taylor expanded in angle around 0 52.5

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

   3. Simplified 52.5

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

   4. Applied egg-rr 53.6

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{0.25 \cdot \left({16}^{0.5} \cdot \left(y-scale \cdot b\right)\right)}\right) \cdot 3}} \]

   5. Applied egg-rr 42.5

      \[\leadsto \color{blue}{{\left({\left(\sqrt[3]{y-scale \cdot b}\right)}^{2}\right)}^{1.5}} \]

   if 2.7985085097999705e-87 < x-scale < 7.5000000000000006e48 or 3.1e173 < x-scale

   1. Initial program 63.2

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

   2. Taylor expanded in x-scale around inf 60.2

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

   3. Simplified 59.5

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

   4. Taylor expanded in y-scale around 0 47.5

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

   5. Simplified 36.9

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

4. Final simplification 39.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -2.3337806862687206 \cdot 10^{-55}:\\ \;\;\;\;\left(-0.25 \cdot \mathsf{hypot}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\\ \mathbf{elif}\;x-scale \leq 2.7985085097999705 \cdot 10^{-87}:\\ \;\;\;\;{\left({\left(\sqrt[3]{b \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{elif}\;x-scale \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;0.25 \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 3.1 \cdot 10^{+173}:\\ \;\;\;\;{\left({\left(\sqrt[3]{b \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	42.4
Cost	46212

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;x-scale \leq -2.3337806862687206 \cdot 10^{-55}:\\ \;\;\;\;\left(-0.25 \cdot \mathsf{hypot}\left(b \cdot \sin t_0, a \cdot \cos t_0\right)\right) \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt[3]{b \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \end{array} \]

Alternative 2
Error	45.9
Cost	26760

\[\begin{array}{l} t_0 := {\left({\left(\sqrt[3]{b \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{if}\;y-scale \leq -1.0354176155531083 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 5.8982577025379025 \cdot 10^{-288}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	45.7
Cost	19784

\[\begin{array}{l} t_0 := {\left({\left(\sqrt[3]{b \cdot y-scale}\right)}^{2}\right)}^{1.5}\\ \mathbf{if}\;y-scale \leq -1.0354176155531083 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 2.2875444094303586 \cdot 10^{-263}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	51.0
Cost	14168

\[\begin{array}{l} \mathbf{if}\;a \leq -2.1851151809986967 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{8} \cdot \left(\left(\sqrt{2} \cdot 0.25\right) \cdot \left(x-scale \cdot a\right)\right)\\ \mathbf{elif}\;a \leq -6.054150808855706 \cdot 10^{-237}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;a \leq 2.141467058566112 \cdot 10^{-251}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot \left(b \cdot y-scale\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.0011576791985715 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{{\left(b \cdot y-scale\right)}^{2}}\\ \mathbf{elif}\;a \leq 1.942254377744882 \cdot 10^{-84}:\\ \;\;\;\;0.25 \cdot \left|y-scale \cdot \sqrt{16 \cdot \left(b \cdot b\right)}\right|\\ \mathbf{elif}\;a \leq 1.3180015806908197 \cdot 10^{-64}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	49.1
Cost	13904

\[\begin{array}{l} t_0 := 0.25 \cdot \left|y-scale \cdot \sqrt{16 \cdot \left(b \cdot b\right)}\right|\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{+153}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;b \leq -4.6714154576622066 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.216112899182976 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{{\left(b \cdot y-scale\right)}^{2}}\\ \mathbf{elif}\;b \leq 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot y-scale\\ \end{array} \]

Alternative 6
Error	47.9
Cost	13904

\[\begin{array}{l} t_0 := 0.25 \cdot \left|y-scale \cdot \sqrt{16 \cdot \left(b \cdot b\right)}\right|\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+147}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;b \leq -6.183907232010621 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.1783862867746225 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{8} \cdot \left(\left(\sqrt{2} \cdot 0.25\right) \cdot \left(x-scale \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot y-scale\\ \end{array} \]

Alternative 7
Error	53.3
Cost	13452

\[\begin{array}{l} t_0 := \sqrt{{\left(b \cdot y-scale\right)}^{2}}\\ \mathbf{if}\;b \leq -2.113571770166177 \cdot 10^{+57}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;b \leq 1.4114488049696395 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{+245}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	53.7
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b \leq -2.113571770166177 \cdot 10^{+57}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+175}:\\ \;\;\;\;0.25 \cdot \sqrt{2 \cdot \left(y-scale \cdot \left(\left(b \cdot y-scale\right) \cdot \left(b \cdot 8\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(b \cdot y-scale + -1\right)\\ \end{array} \]

Alternative 9
Error	54.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.1144293867127053 \cdot 10^{-172}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;angle \leq -1.129192539515017 \cdot 10^{-257}:\\ \;\;\;\;0.25 \cdot \sqrt{y-scale \cdot \left(16 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot y-scale\\ \end{array} \]

Alternative 10
Error	54.5
Cost	192

\[b \cdot y-scale \]

Reproduce

herbie shell --seed 2022241 
(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))))