Average Error: 63.9 → 41.9
Time: 2.5min
Precision: binary64
Cost: 46484
\[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
↓
\[\begin{array}{l} t_0 := a \cdot \sqrt{8}\\ t_1 := \left(0.25 \cdot x-scale\right) \cdot \left(\left|t_0\right| \cdot \sqrt{2}\right)\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot t_0\right)\right) \cdot \sqrt{0}\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(e^{\mathsf{log1p}\left(b \cdot \sqrt{8}\right)} + -1\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+236}:\\ \;\;\;\;\sqrt[3]{{\left(a \cdot \sqrt{1 + {\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2}}\right)}^{3}} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
(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 (* a (sqrt 8.0)))
        (t_1 (* (* 0.25 x-scale) (* (fabs t_0) (sqrt 2.0)))))
   (if (<= b -1.1e+127)
     t_1
     (if (<= b -1.15e-47)
       (* (* 0.25 (* (* x-scale y-scale) t_0)) (sqrt 0.0))
       (if (<= b -1.05e-111)
         t_1
         (if (<= b 1.35e-43)
           (*
            0.25
            (*
             (* x-scale (+ (exp (log1p (* b (sqrt 8.0)))) -1.0))
             (sin (* PI (* 0.005555555555555556 angle)))))
           (if (<= b 1.85e+236)
             (*
              (cbrt
               (pow
                (*
                 a
                 (sqrt
                  (+
                   1.0
                   (pow (cos (* 0.005555555555555556 (* PI angle))) 2.0))))
                3.0))
              (* (* x-scale (sqrt 8.0)) -0.25))
             (* x-scale a))))))))
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 = a * sqrt(8.0);
	double t_1 = (0.25 * x_45_scale) * (fabs(t_0) * sqrt(2.0));
	double tmp;
	if (b <= -1.1e+127) {
		tmp = t_1;
	} else if (b <= -1.15e-47) {
		tmp = (0.25 * ((x_45_scale * y_45_scale) * t_0)) * sqrt(0.0);
	} else if (b <= -1.05e-111) {
		tmp = t_1;
	} else if (b <= 1.35e-43) {
		tmp = 0.25 * ((x_45_scale * (exp(log1p((b * sqrt(8.0)))) + -1.0)) * sin((((double) M_PI) * (0.005555555555555556 * angle))));
	} else if (b <= 1.85e+236) {
		tmp = cbrt(pow((a * sqrt((1.0 + pow(cos((0.005555555555555556 * (((double) M_PI) * angle))), 2.0)))), 3.0)) * ((x_45_scale * sqrt(8.0)) * -0.25);
	} else {
		tmp = x_45_scale * a;
	}
	return tmp;
}

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 = a * Math.sqrt(8.0);
	double t_1 = (0.25 * x_45_scale) * (Math.abs(t_0) * Math.sqrt(2.0));
	double tmp;
	if (b <= -1.1e+127) {
		tmp = t_1;
	} else if (b <= -1.15e-47) {
		tmp = (0.25 * ((x_45_scale * y_45_scale) * t_0)) * Math.sqrt(0.0);
	} else if (b <= -1.05e-111) {
		tmp = t_1;
	} else if (b <= 1.35e-43) {
		tmp = 0.25 * ((x_45_scale * (Math.exp(Math.log1p((b * Math.sqrt(8.0)))) + -1.0)) * Math.sin((Math.PI * (0.005555555555555556 * angle))));
	} else if (b <= 1.85e+236) {
		tmp = Math.cbrt(Math.pow((a * Math.sqrt((1.0 + Math.pow(Math.cos((0.005555555555555556 * (Math.PI * angle))), 2.0)))), 3.0)) * ((x_45_scale * Math.sqrt(8.0)) * -0.25);
	} else {
		tmp = x_45_scale * a;
	}
	return tmp;
}

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(a * sqrt(8.0))
	t_1 = Float64(Float64(0.25 * x_45_scale) * Float64(abs(t_0) * sqrt(2.0)))
	tmp = 0.0
	if (b <= -1.1e+127)
		tmp = t_1;
	elseif (b <= -1.15e-47)
		tmp = Float64(Float64(0.25 * Float64(Float64(x_45_scale * y_45_scale) * t_0)) * sqrt(0.0));
	elseif (b <= -1.05e-111)
		tmp = t_1;
	elseif (b <= 1.35e-43)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale * Float64(exp(log1p(Float64(b * sqrt(8.0)))) + -1.0)) * sin(Float64(pi * Float64(0.005555555555555556 * angle)))));
	elseif (b <= 1.85e+236)
		tmp = Float64(cbrt((Float64(a * sqrt(Float64(1.0 + (cos(Float64(0.005555555555555556 * Float64(pi * angle))) ^ 2.0)))) ^ 3.0)) * Float64(Float64(x_45_scale * sqrt(8.0)) * -0.25));
	else
		tmp = Float64(x_45_scale * a);
	end
	return tmp
end

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[(a * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.25 * x$45$scale), $MachinePrecision] * N[(N[Abs[t$95$0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1e+127], t$95$1, If[LessEqual[b, -1.15e-47], N[(N[(0.25 * N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.05e-111], t$95$1, If[LessEqual[b, 1.35e-43], N[(0.25 * N[(N[(x$45$scale * N[(N[Exp[N[Log[1 + N[(b * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.85e+236], N[(N[Power[N[Power[N[(a * N[Sqrt[N[(1.0 + N[Power[N[Cos[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(x$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], N[(x$45$scale * a), $MachinePrecision]]]]]]]]

\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 := a \cdot \sqrt{8}\\
t_1 := \left(0.25 \cdot x-scale\right) \cdot \left(\left|t_0\right| \cdot \sqrt{2}\right)\\
\mathbf{if}\;b \leq -1.1 \cdot 10^{+127}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;b \leq -1.15 \cdot 10^{-47}:\\
\;\;\;\;\left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot t_0\right)\right) \cdot \sqrt{0}\\

\mathbf{elif}\;b \leq -1.05 \cdot 10^{-111}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;b \leq 1.35 \cdot 10^{-43}:\\
\;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(e^{\mathsf{log1p}\left(b \cdot \sqrt{8}\right)} + -1\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\

\mathbf{elif}\;b \leq 1.85 \cdot 10^{+236}:\\
\;\;\;\;\sqrt[3]{{\left(a \cdot \sqrt{1 + {\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2}}\right)}^{3}} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot -0.25\right)\\

\mathbf{else}:\\
\;\;\;\;x-scale \cdot a\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if b < -1.1000000000000001e127 or -1.14999999999999991e-47 < b < -1.0499999999999999e-111

    1. Initial program 63.9

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

    2. Simplified63.7

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

      [Start]63.9

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

    3. Taylor expanded in angle around 0 47.5

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

    4. Simplified47.5

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

      [Start]47.5

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

      associate-*r* [=>]47.5

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

      *-commutative [=>]47.5

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

      *-commutative [=>]47.5

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

    5. Applied egg-rr47.6

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

    6. Simplified47.0

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

      [Start]47.6

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

      rem-square-sqrt [<=]47.6

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

      swap-sqr [<=]47.6

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

      rem-sqrt-square [=>]47.0

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

      *-commutative [<=]47.0

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

    if -1.1000000000000001e127 < b < -1.14999999999999991e-47

    1. Initial program 63.9

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

    2. Simplified62.9

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

      [Start]63.9

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

    3. Taylor expanded in a around inf 62.4

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

    4. Simplified62.4

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

      [Start]62.4

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

      associate-*r* [=>]62.4

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

      associate-*r* [=>]62.5

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

      *-commutative [=>]62.5

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

      *-commutative [=>]62.5

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

    5. Taylor expanded in y-scale around 0 58.2

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

    6. Simplified58.4

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

      [Start]58.2

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

      unpow2 [=>]58.2

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

      associate-*r* [=>]59.0

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

      *-commutative [=>]59.0

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

      *-commutative [=>]59.0

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

    7. Taylor expanded in angle around 0 51.4

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

    if -1.0499999999999999e-111 < b < 1.34999999999999996e-43

    1. Initial program 64.0

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

    2. Simplified62.8

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

      [Start]64.0

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

    3. Taylor expanded in angle around 0 61.1

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

    4. Simplified61.1

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

      [Start]61.1

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

      unpow2 [=>]61.1

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

      unpow2 [=>]61.1

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

    5. Taylor expanded in b around inf 41.0

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

    6. Simplified42.7

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

      [Start]41.0

      \[ 0.25 \cdot \left(x-scale \cdot \left(b \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)\right) \]

      associate-*r* [=>]41.0

      \[ \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(b \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{8}\right)\right)} \]

      *-commutative [<=]41.0

      \[ \left(0.25 \cdot x-scale\right) \cdot \left(b \cdot \color{blue}{\left(\sqrt{8} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \]

      associate-*r* [<=]41.0

      \[ \color{blue}{0.25 \cdot \left(x-scale \cdot \left(b \cdot \left(\sqrt{8} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)} \]

      associate-*r* [=>]41.0

      \[ 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(\left(b \cdot \sqrt{8}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \]

      associate-*r* [=>]42.7

      \[ 0.25 \cdot \color{blue}{\left(\left(x-scale \cdot \left(b \cdot \sqrt{8}\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      *-commutative [=>]42.7

      \[ 0.25 \cdot \left(\left(x-scale \cdot \color{blue}{\left(\sqrt{8} \cdot b\right)}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      associate-*r* [=>]42.7

      \[ 0.25 \cdot \left(\left(x-scale \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \sin \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right) \]

      *-commutative [=>]42.7

      \[ 0.25 \cdot \left(\left(x-scale \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \]

    7. Applied egg-rr31.2

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

    if 1.34999999999999996e-43 < b < 1.85000000000000007e236

    1. Initial program 63.9

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

    2. Simplified63.3

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

      [Start]63.9

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

    3. Taylor expanded in angle around 0 60.7

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

    4. Simplified60.7

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

      [Start]60.7

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

      unpow2 [=>]60.7

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

      unpow2 [=>]60.7

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

    5. Taylor expanded in y-scale around 0 58.9

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

    6. Simplified58.9

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

      [Start]58.9

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

      associate-*r* [=>]58.9

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

      +-commutative [<=]58.9

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

      *-commutative [=>]58.9

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

    7. Taylor expanded in a around -inf 47.5

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

    8. Applied egg-rr48.5

      \[\leadsto \left(-1 \cdot \color{blue}{\sqrt[3]{{\left(a \cdot \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + 1}\right)}^{3}}}\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \]

    if 1.85000000000000007e236 < b

    1. Initial program 64.0

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

    2. Simplified64.0

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

      [Start]64.0

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

    3. Taylor expanded in angle around 0 45.3

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

    4. Simplified45.3

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

      [Start]45.3

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

      associate-*r* [=>]45.3

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

      *-commutative [=>]45.3

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

      *-commutative [=>]45.3

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

    5. Applied egg-rr46.9

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

    6. Simplified46.9

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

      [Start]46.9

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

      associate-*r* [=>]46.9

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

      *-commutative [=>]46.9

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

      metadata-eval [=>]46.9

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

    7. Applied egg-rr48.3

      \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{\left(1 + \left(a \cdot a\right) \cdot 16\right) - 1}} \]

    8. Taylor expanded in x-scale around 0 45.2

      \[\leadsto \color{blue}{x-scale \cdot a} \]
  3. Recombined 5 regimes into one program.

  4. Final simplification41.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+127}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(\left|a \cdot \sqrt{8}\right| \cdot \sqrt{2}\right)\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{0}\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-111}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(\left|a \cdot \sqrt{8}\right| \cdot \sqrt{2}\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(e^{\mathsf{log1p}\left(b \cdot \sqrt{8}\right)} + -1\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+236}:\\ \;\;\;\;\sqrt[3]{{\left(a \cdot \sqrt{1 + {\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2}}\right)}^{3}} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]

Alternatives

Alternative 1
Error41.5
Cost33424
\[\begin{array}{l} t_0 := a \cdot \sqrt{8}\\ t_1 := \left(0.25 \cdot x-scale\right) \cdot \left(\left|t_0\right| \cdot \sqrt{2}\right)\\ \mathbf{if}\;b \leq -7.7 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.95 \cdot 10^{-47}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot t_0\right)\right) \cdot \sqrt{0}\\ \mathbf{elif}\;b \leq -1.28 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-20}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(e^{\mathsf{log1p}\left(b \cdot \sqrt{8}\right)} + -1\right)\right) \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
Alternative 2
Error42.7
Cost20624
\[\begin{array}{l} t_0 := a \cdot \sqrt{8}\\ t_1 := \left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot t_0\right)\right) \cdot \sqrt{0}\\ t_2 := \left(0.25 \cdot x-scale\right) \cdot \left(\left|t_0\right| \cdot \sqrt{2}\right)\\ \mathbf{if}\;b \leq -5.3 \cdot 10^{+126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-133}:\\ \;\;\;\;0.25 \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(x-scale \cdot \sqrt{8 \cdot \left(b \cdot b\right)}\right)\right)\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
Alternative 3
Error41.8
Cost19908
\[\begin{array}{l} t_0 := a \cdot \sqrt{8}\\ \mathbf{if}\;b \leq -2.7 \cdot 10^{+133}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \left(\left|t_0\right| \cdot \sqrt{2}\right)\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{+79}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot t_0\right)\right) \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
Alternative 4
Error41.8
Cost19780
\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+132}:\\ \;\;\;\;\left(4 \cdot \left|a\right|\right) \cdot \left(x-scale \cdot \log \left(e^{0.25}\right)\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+72}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
Alternative 5
Error41.0
Cost13769
\[\begin{array}{l} \mathbf{if}\;y-scale \leq -1.45 \cdot 10^{-62} \lor \neg \left(y-scale \leq 2400\right):\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(x-scale \cdot y-scale\right) \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{0}\\ \end{array} \]
Alternative 6
Error45.5
Cost7497
\[\begin{array}{l} \mathbf{if}\;y-scale \leq -4.4 \cdot 10^{+93} \lor \neg \left(y-scale \leq 3.2 \cdot 10^{-53}\right):\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \sqrt{-1 + \left(1 + \left(a \cdot a\right) \cdot 16\right)}\\ \end{array} \]
Alternative 7
Error47.1
Cost7240
\[\begin{array}{l} \mathbf{if}\;y-scale \leq -4.2 \cdot 10^{-115}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;y-scale \leq 6.1 \cdot 10^{-51}:\\ \;\;\;\;\left(0.25 \cdot x-scale\right) \cdot \sqrt{\left(a \cdot a\right) \cdot 16}\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
Alternative 8
Error49.2
Cost521
\[\begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-109} \lor \neg \left(b \leq 2.2 \cdot 10^{+247}\right):\\ \;\;\;\;x-scale \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-x-scale\right)\\ \end{array} \]
Alternative 9
Error49.2
Cost192
\[x-scale \cdot a \]

Error

Reproduce

herbie shell --seed 2023011 
(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))))