a from scale-rotated-ellipse

Average Accuracy: 0.8% → 29.7%

Time: 1.7min

Precision: binary64

Cost: 59597

Math

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

↓

\[\begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;y-scale \leq -6.2 \cdot 10^{+87}:\\ \;\;\;\;\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2 \cdot \left({\left(\cos t_0 \cdot b\right)}^{2} + {\left(a \cdot \sin t_0\right)}^{2}\right)} \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq -3.5 \cdot 10^{-200} \lor \neg \left(y-scale \leq 1.92 \cdot 10^{-147}\right):\\ \;\;\;\;0.25 \cdot \left(4 \cdot \left|y-scale \cdot b\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos t_1}^{2}, {\sin t_1}^{2} \cdot \left(b \cdot b\right)\right)}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle)))
        (t_1 (* 0.005555555555555556 (* PI angle))))
   (if (<= y-scale -6.2e+87)
     (*
      (* y-scale (sqrt 8.0))
      (*
       (sqrt (* 2.0 (+ (pow (* (cos t_0) b) 2.0) (pow (* a (sin t_0)) 2.0))))
       -0.25))
     (if (or (<= y-scale -3.5e-200) (not (<= y-scale 1.92e-147)))
       (* 0.25 (* 4.0 (fabs (* y-scale b))))
       (*
        0.25
        (*
         x-scale
         (*
          (sqrt 8.0)
          (sqrt
           (*
            2.0
            (fma
             (* a a)
             (pow (cos t_1) 2.0)
             (* (pow (sin t_1) 2.0) (* b b))))))))))))

C

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

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_1 = 0.005555555555555556 * (((double) M_PI) * angle);
	double tmp;
	if (y_45_scale <= -6.2e+87) {
		tmp = (y_45_scale * sqrt(8.0)) * (sqrt((2.0 * (pow((cos(t_0) * b), 2.0) + pow((a * sin(t_0)), 2.0)))) * -0.25);
	} else if ((y_45_scale <= -3.5e-200) || !(y_45_scale <= 1.92e-147)) {
		tmp = 0.25 * (4.0 * fabs((y_45_scale * b)));
	} else {
		tmp = 0.25 * (x_45_scale * (sqrt(8.0) * sqrt((2.0 * fma((a * a), pow(cos(t_1), 2.0), (pow(sin(t_1), 2.0) * (b * b)))))));
	}
	return tmp;
}

Julia

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

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_1 = Float64(0.005555555555555556 * Float64(pi * angle))
	tmp = 0.0
	if (y_45_scale <= -6.2e+87)
		tmp = Float64(Float64(y_45_scale * sqrt(8.0)) * Float64(sqrt(Float64(2.0 * Float64((Float64(cos(t_0) * b) ^ 2.0) + (Float64(a * sin(t_0)) ^ 2.0)))) * -0.25));
	elseif ((y_45_scale <= -3.5e-200) || !(y_45_scale <= 1.92e-147))
		tmp = Float64(0.25 * Float64(4.0 * abs(Float64(y_45_scale * b))));
	else
		tmp = Float64(0.25 * Float64(x_45_scale * Float64(sqrt(8.0) * sqrt(Float64(2.0 * fma(Float64(a * a), (cos(t_1) ^ 2.0), Float64((sin(t_1) ^ 2.0) * Float64(b * b))))))));
	end
	return tmp
end

Wolfram

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

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -6.2e+87], N[(N[(y$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(N[Power[N[(N[Cos[t$95$0], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$45$scale, -3.5e-200], N[Not[LessEqual[y$45$scale, 1.92e-147]], $MachinePrecision]], N[(0.25 * N[(4.0 * N[Abs[N[(y$45$scale * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(x$45$scale * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(a * a), $MachinePrecision] * N[Power[N[Cos[t$95$1], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < -6.1999999999999999e87

   1. Initial program 0.4%

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

   2. Simplified 1.3%

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

      [Start] 0.4

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

   3. Taylor expanded in y-scale around inf 4.6%

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

   4. Taylor expanded in x-scale around -inf 37.9%

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

   5. Simplified 43.7%

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

      [Start] 37.9	\[ -0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right) \]
      *-commutative [=>] 37.9	\[ \color{blue}{\left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)}\right) \cdot -0.25} \]
      associate-*l* [=>] 37.9	\[ \color{blue}{\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}\right)} \cdot -0.25\right)} \]

   if -6.1999999999999999e87 < y-scale < -3.50000000000000023e-200 or 1.9200000000000001e-147 < y-scale

   1. Initial program 1.1%

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

   2. Taylor expanded in angle around 0 16.1%

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

   3. Applied egg-rr 31.1%

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

      [Start] 16.1	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \]
      add-sqr-sqrt [=>] 15.3	\[ 0.25 \cdot \color{blue}{\left(\sqrt{\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)} \cdot \sqrt{\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)}\right)} \]
      sqrt-unprod [=>] 18.4	\[ 0.25 \cdot \color{blue}{\sqrt{\left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right)}} \]
      rem-sqrt-square [=>] 30.8	\[ 0.25 \cdot \color{blue}{\left|\sqrt{2} \cdot \left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right)\right|} \]
      *-commutative [=>] 30.8	\[ 0.25 \cdot \left|\color{blue}{\left(y-scale \cdot \left(b \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}}\right| \]
      associate-*r* [=>] 30.7	\[ 0.25 \cdot \left|\color{blue}{\left(\left(y-scale \cdot b\right) \cdot \sqrt{8}\right)} \cdot \sqrt{2}\right| \]
      associate-*l* [=>] 30.6	\[ 0.25 \cdot \left|\color{blue}{\left(y-scale \cdot b\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)}\right| \]
      *-commutative [=>] 30.6	\[ 0.25 \cdot \left|\color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot \left(y-scale \cdot b\right)}\right| \]
      sqrt-unprod [=>] 31.1	\[ 0.25 \cdot \left|\color{blue}{\sqrt{8 \cdot 2}} \cdot \left(y-scale \cdot b\right)\right| \]
      metadata-eval [=>] 31.1	\[ 0.25 \cdot \left|\sqrt{\color{blue}{16}} \cdot \left(y-scale \cdot b\right)\right| \]
      metadata-eval [=>] 31.1	\[ 0.25 \cdot \left|\color{blue}{4} \cdot \left(y-scale \cdot b\right)\right| \]

   if -3.50000000000000023e-200 < y-scale < 1.9200000000000001e-147

   1. Initial program 0.1%

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

   2. Simplified 0.0%

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

      [Start] 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 y-scale around 0 19.1%

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

   4. Simplified 19.1%

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

      [Start] 19.1	\[ 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right) \]
      associate-*l* [=>] 19.1	\[ 0.25 \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \]
      distribute-lft-out [=>] 19.1	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      fma-def [=>] 19.1	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left({a}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right)\right) \]
      unpow2 [=>] 19.1	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(\color{blue}{a \cdot a}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right) \]
      *-commutative [=>] 19.1	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \color{blue}{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {b}^{2}}\right)}\right)\right) \]
      unpow2 [=>] 19.1	\[ 0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \color{blue}{\left(b \cdot b\right)}\right)}\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -6.2 \cdot 10^{+87}:\\ \;\;\;\;\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2 \cdot \left({\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot b\right)}^{2} + {\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\right)} \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq -3.5 \cdot 10^{-200} \lor \neg \left(y-scale \leq 1.92 \cdot 10^{-147}\right):\\ \;\;\;\;0.25 \cdot \left(4 \cdot \left|y-scale \cdot b\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	29.2%
Cost	52804

\[\begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;y-scale \leq -5.2 \cdot 10^{+89}:\\ \;\;\;\;\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2 \cdot \left({\left(\cos t_0 \cdot b\right)}^{2} + {\left(a \cdot \sin t_0\right)}^{2}\right)} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(4 \cdot \left|y-scale \cdot b\right|\right)\\ \end{array} \]

Alternative 2
Accuracy	28.5%
Cost	6848

\[0.25 \cdot \left(4 \cdot \left|y-scale \cdot b\right|\right) \]

Alternative 3
Accuracy	15.5%
Cost	448

\[0.25 \cdot \left(4 \cdot \left(y-scale \cdot b\right)\right) \]

Alternative 4
Accuracy	4.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023135 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))