?

Average Accuracy: 0.1% → 36.8%
Time: 2.5min
Precision: binary64
Cost: 19844

?

\[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+69}:\\ \;\;\;\;0.25 \cdot \sqrt{{\left(\sqrt[3]{x-scale \cdot a}\right)}^{6} \cdot 16}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+61}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right) \cdot \sqrt{0}\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \end{array} \]
(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
 (if (<= b -1.5e+69)
   (* 0.25 (sqrt (* (pow (cbrt (* x-scale a)) 6.0) 16.0)))
   (if (<= b 6.2e+61)
     (* 0.25 (* (* y-scale (* b (sqrt 8.0))) (sqrt 0.0)))
     (* 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 tmp;
	if (b <= -1.5e+69) {
		tmp = 0.25 * sqrt((pow(cbrt((x_45_scale * a)), 6.0) * 16.0));
	} else if (b <= 6.2e+61) {
		tmp = 0.25 * ((y_45_scale * (b * sqrt(8.0))) * sqrt(0.0));
	} 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 tmp;
	if (b <= -1.5e+69) {
		tmp = 0.25 * Math.sqrt((Math.pow(Math.cbrt((x_45_scale * a)), 6.0) * 16.0));
	} else if (b <= 6.2e+61) {
		tmp = 0.25 * ((y_45_scale * (b * Math.sqrt(8.0))) * Math.sqrt(0.0));
	} 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)
	tmp = 0.0
	if (b <= -1.5e+69)
		tmp = Float64(0.25 * sqrt(Float64((cbrt(Float64(x_45_scale * a)) ^ 6.0) * 16.0)));
	elseif (b <= 6.2e+61)
		tmp = Float64(0.25 * Float64(Float64(y_45_scale * Float64(b * sqrt(8.0))) * sqrt(0.0)));
	else
		tmp = Float64(x_45_scale * Float64(-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_] := If[LessEqual[b, -1.5e+69], N[(0.25 * N[Sqrt[N[(N[Power[N[Power[N[(x$45$scale * a), $MachinePrecision], 1/3], $MachinePrecision], 6.0], $MachinePrecision] * 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e+61], N[(0.25 * N[(N[(y$45$scale * N[(b * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $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}
\mathbf{if}\;b \leq -1.5 \cdot 10^{+69}:\\
\;\;\;\;0.25 \cdot \sqrt{{\left(\sqrt[3]{x-scale \cdot a}\right)}^{6} \cdot 16}\\

\mathbf{elif}\;b \leq 6.2 \cdot 10^{+61}:\\
\;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right) \cdot \sqrt{0}\right)\\

\mathbf{else}:\\
\;\;\;\;x-scale \cdot \left(-a\right)\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 3 regimes

  2. if b < -1.49999999999999992e69

    1. Initial program 0.0%

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

    2. Simplified0.1%

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

      [Start]0.0

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

    3. Taylor expanded in angle around 0 3.8%

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

    4. Simplified3.8%

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

      [Start]3.8

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

      associate-*r/ [=>]3.8

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

      unpow2 [=>]3.8

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

      unpow2 [=>]3.8

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

    5. Applied egg-rr6.4%

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

    6. Taylor expanded in x-scale around 0 28.8%

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

    7. Simplified28.8%

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

      [Start]28.8

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

      *-commutative [=>]28.8

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

      associate-*r* [<=]28.8

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

      associate-*l* [=>]28.8

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

    8. Applied egg-rr34.3%

      \[\leadsto 0.25 \cdot \color{blue}{{\left(2 \cdot {\left(a \cdot \left(\sqrt{8} \cdot x-scale\right)\right)}^{2}\right)}^{0.5}} \]

    9. Simplified34.1%

      \[\leadsto 0.25 \cdot \color{blue}{\sqrt{{\left(\sqrt[3]{x-scale \cdot a}\right)}^{6} \cdot 16}} \]
      Proof

      [Start]34.3

      \[ 0.25 \cdot {\left(2 \cdot {\left(a \cdot \left(\sqrt{8} \cdot x-scale\right)\right)}^{2}\right)}^{0.5} \]

      unpow1/2 [=>]34.3

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

      *-commutative [=>]34.3

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

      unpow2 [=>]34.3

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

      swap-sqr [=>]26.4

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

      *-commutative [=>]26.4

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

      *-commutative [=>]26.4

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

      swap-sqr [=>]26.4

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

      associate-*l* [<=]26.4

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

      swap-sqr [<=]34.2

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

      associate-*l* [=>]34.2

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

      rem-cube-cbrt [<=]34.2

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

      rem-cube-cbrt [<=]34.1

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

      pow-sqr [=>]34.1

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

      *-commutative [=>]34.1

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

      metadata-eval [=>]34.1

      \[ 0.25 \cdot \sqrt{{\left(\sqrt[3]{x-scale \cdot a}\right)}^{\color{blue}{6}} \cdot \left(\left(\sqrt{8} \cdot \sqrt{8}\right) \cdot 2\right)} \]

      rem-square-sqrt [=>]34.1

      \[ 0.25 \cdot \sqrt{{\left(\sqrt[3]{x-scale \cdot a}\right)}^{6} \cdot \left(\color{blue}{8} \cdot 2\right)} \]

      metadata-eval [=>]34.1

      \[ 0.25 \cdot \sqrt{{\left(\sqrt[3]{x-scale \cdot a}\right)}^{6} \cdot \color{blue}{16}} \]

    if -1.49999999999999992e69 < b < 6.1999999999999998e61

    1. Initial program 0.1%

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

    2. Simplified2.1%

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

      [Start]0.1

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

    3. Taylor expanded in b around inf 1.6%

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

    4. Simplified1.6%

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

      [Start]1.6

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

      unpow2 [=>]1.6

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

      associate-*l* [=>]1.6

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

    5. Taylor expanded in x-scale around inf 10.9%

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

    6. Simplified9.8%

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

      [Start]10.9

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

    7. Taylor expanded in angle around 0 40.8%

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

    if 6.1999999999999998e61 < b

    1. Initial program 0.0%

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

    2. Simplified0.2%

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

      [Start]0.0

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

    3. Taylor expanded in angle around 0 4.3%

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

    4. Simplified4.3%

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

      [Start]4.3

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

      unpow2 [=>]4.3

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

      unpow2 [=>]4.3

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

    5. Taylor expanded in x-scale around -inf 27.4%

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

    6. Simplified27.4%

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

      [Start]27.4

      \[ -1 \cdot \left(x-scale \cdot a\right) \]

      mul-1-neg [=>]27.4

      \[ \color{blue}{-x-scale \cdot a} \]

      distribute-rgt-neg-in [=>]27.4

      \[ \color{blue}{x-scale \cdot \left(-a\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification36.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+69}:\\ \;\;\;\;0.25 \cdot \sqrt{{\left(\sqrt[3]{x-scale \cdot a}\right)}^{6} \cdot 16}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+61}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right) \cdot \sqrt{0}\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy26.3%
Cost13708
\[\begin{array}{l} t_0 := {\left(x-scale \cdot y-scale\right)}^{2}\\ t_1 := x-scale \cdot \left(-a\right)\\ \mathbf{if}\;y-scale \leq -3.8 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{a \cdot \frac{t_0}{y-scale \cdot y-scale}}{x-scale}\\ \mathbf{elif}\;y-scale \leq -1.75 \cdot 10^{-290}:\\ \;\;\;\;-0.25 \cdot \sqrt[3]{{\left(x-scale \cdot \left(a \cdot 4\right)\right)}^{3}}\\ \mathbf{elif}\;y-scale \leq 2.4 \cdot 10^{-113}:\\ \;\;\;\;\frac{t_0 \cdot \frac{\frac{a}{x-scale}}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Accuracy27.9%
Cost13641
\[\begin{array}{l} \mathbf{if}\;angle \leq -8.5 \cdot 10^{-30} \lor \neg \left(angle \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;-0.25 \cdot \left(e^{\mathsf{log1p}\left(x-scale \cdot \left(a \cdot 4\right)\right)} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \end{array} \]
Alternative 3
Accuracy32.8%
Cost13641
\[\begin{array}{l} \mathbf{if}\;angle \leq -1.8 \cdot 10^{-54} \lor \neg \left(angle \leq 9.2 \cdot 10^{-32}\right):\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right) \cdot \sqrt{0}\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \end{array} \]
Alternative 4
Accuracy23.2%
Cost7696
\[\begin{array}{l} t_0 := {\left(x-scale \cdot y-scale\right)}^{2}\\ t_1 := x-scale \cdot \left(-a\right)\\ \mathbf{if}\;y-scale \leq -1.95 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{a \cdot \frac{t_0}{y-scale \cdot y-scale}}{x-scale}\\ \mathbf{elif}\;y-scale \leq -3.8 \cdot 10^{-287}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;y-scale \leq 8.8 \cdot 10^{-128}:\\ \;\;\;\;\frac{a \cdot \frac{t_0}{x-scale \cdot y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Accuracy25.2%
Cost7696
\[\begin{array}{l} t_0 := {\left(x-scale \cdot y-scale\right)}^{2}\\ t_1 := x-scale \cdot \left(-a\right)\\ \mathbf{if}\;y-scale \leq -5 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y-scale \leq -2.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{a \cdot \frac{t_0}{y-scale \cdot y-scale}}{x-scale}\\ \mathbf{elif}\;y-scale \leq -8.2 \cdot 10^{-287}:\\ \;\;\;\;x-scale \cdot a\\ \mathbf{elif}\;y-scale \leq 6 \cdot 10^{-124}:\\ \;\;\;\;\frac{t_0 \cdot \frac{\frac{a}{x-scale}}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Accuracy22.9%
Cost7433
\[\begin{array}{l} \mathbf{if}\;y-scale \leq -7.6 \cdot 10^{-287} \lor \neg \left(y-scale \leq 1.9 \cdot 10^{-124}\right):\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{\left(x-scale \cdot y-scale\right)}^{2}}{x-scale \cdot y-scale}}{y-scale}\\ \end{array} \]
Alternative 7
Accuracy23.6%
Cost1225
\[\begin{array}{l} \mathbf{if}\;y-scale \leq -1.2 \cdot 10^{-142} \lor \neg \left(y-scale \leq 5.5 \cdot 10^{-140}\right):\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \frac{x-scale \cdot \left(y-scale \cdot y-scale\right)}{y-scale \cdot \frac{x-scale}{\frac{a}{y-scale}}}\\ \end{array} \]
Alternative 8
Accuracy22.7%
Cost521
\[\begin{array}{l} \mathbf{if}\;x-scale \leq 2 \cdot 10^{-12} \lor \neg \left(x-scale \leq 8.5 \cdot 10^{+298}\right):\\ \;\;\;\;x-scale \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
Alternative 9
Accuracy22.5%
Cost192
\[x-scale \cdot a \]

Error

Reproduce?

herbie shell --seed 2023125 
(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))))