a from scale-rotated-ellipse

Average Error: 63.5 → 38.1

Time: 2.4min

Precision: binary64

Cost: 46344

Math

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

↓

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;x-scale \leq -1.7371761881782182 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right) \cdot \left(\left(x-scale \cdot {16}^{0.5}\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq 2.730233269999451 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot e^{\log \left(\sqrt{{\left(\sqrt[3]{4 \cdot \left(b \cdot y-scale\right)}\right)}^{2}}\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(a \cdot \cos t_0, b \cdot \sin t_0\right) \cdot \left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\\ \end{array} \]

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 (* 0.005555555555555556 (* PI angle))))
   (if (<= x-scale -1.7371761881782182e-59)
     (*
      (hypot (* b (sin (* (* 0.005555555555555556 PI) angle))) a)
      (* (* x-scale (pow 16.0 0.5)) -0.25))
     (if (<= x-scale 2.730233269999451e-27)
       (*
        0.25
        (exp (* (log (sqrt (pow (cbrt (* 4.0 (* b y-scale))) 2.0))) 3.0)))
       (*
        (hypot (* a (cos t_0)) (* b (sin t_0)))
        (* (* 0.25 (sqrt 2.0)) (* x-scale (sqrt 8.0))))))))

C

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

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	double tmp;
	if (x_45_scale <= -1.7371761881782182e-59) {
		tmp = hypot((b * sin(((0.005555555555555556 * ((double) M_PI)) * angle))), a) * ((x_45_scale * pow(16.0, 0.5)) * -0.25);
	} else if (x_45_scale <= 2.730233269999451e-27) {
		tmp = 0.25 * exp((log(sqrt(pow(cbrt((4.0 * (b * y_45_scale))), 2.0))) * 3.0));
	} else {
		tmp = hypot((a * cos(t_0)), (b * sin(t_0))) * ((0.25 * sqrt(2.0)) * (x_45_scale * sqrt(8.0)));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (Math.PI * angle);
	double tmp;
	if (x_45_scale <= -1.7371761881782182e-59) {
		tmp = Math.hypot((b * Math.sin(((0.005555555555555556 * Math.PI) * angle))), a) * ((x_45_scale * Math.pow(16.0, 0.5)) * -0.25);
	} else if (x_45_scale <= 2.730233269999451e-27) {
		tmp = 0.25 * Math.exp((Math.log(Math.sqrt(Math.pow(Math.cbrt((4.0 * (b * y_45_scale))), 2.0))) * 3.0));
	} else {
		tmp = Math.hypot((a * Math.cos(t_0)), (b * Math.sin(t_0))) * ((0.25 * Math.sqrt(2.0)) * (x_45_scale * Math.sqrt(8.0)));
	}
	return tmp;
}

Julia

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

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	tmp = 0.0
	if (x_45_scale <= -1.7371761881782182e-59)
		tmp = Float64(hypot(Float64(b * sin(Float64(Float64(0.005555555555555556 * pi) * angle))), a) * Float64(Float64(x_45_scale * (16.0 ^ 0.5)) * -0.25));
	elseif (x_45_scale <= 2.730233269999451e-27)
		tmp = Float64(0.25 * exp(Float64(log(sqrt((cbrt(Float64(4.0 * Float64(b * y_45_scale))) ^ 2.0))) * 3.0)));
	else
		tmp = Float64(hypot(Float64(a * cos(t_0)), Float64(b * sin(t_0))) * Float64(Float64(0.25 * sqrt(2.0)) * Float64(x_45_scale * sqrt(8.0))));
	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[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.7371761881782182e-59], N[(N[Sqrt[N[(b * N[Sin[N[(N[(0.005555555555555556 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + a ^ 2], $MachinePrecision] * N[(N[(x$45$scale * N[Power[16.0, 0.5], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2.730233269999451e-27], N[(0.25 * N[Exp[N[(N[Log[N[Sqrt[N[Power[N[Power[N[(4.0 * N[(b * y$45$scale), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(0.25 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -1.73717618817821823e-59

   1. Initial program 63.4

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

   2. Taylor expanded in x-scale around inf 60.0

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

   3. Simplified 59.2

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale} \cdot \frac{a \cdot a}{y-scale} + \frac{b \cdot b}{y-scale} \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale}\right)}} \]
      Proof
      (*.f64 (*.f64 1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (*.f64 a a) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 x-scale y-scale) (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (*.f64 a a) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 18 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 y-scale x-scale)) (sqrt.f64 8))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (*.f64 a a) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (Rewrite<= associate-*r*_binary64 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8))))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (*.f64 a a) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 11 points increase in error, 15 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 a 2)) (*.f64 y-scale y-scale))) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 22 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.f64 y-scale y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2))) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2)) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 y-scale y-scale))))))): 2 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2)) (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2))) (*.f64 2 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 1/4 (*.f64 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 2 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2))) (*.f64 2 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2)))))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in y-scale around -inf 46.1

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

   5. Simplified 35.4

      \[\leadsto \color{blue}{\mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a \cdot \cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot -0.25\right)} \]
      Proof
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (*.f64 (*.f64 1/180 (PI.f64)) angle))) (*.f64 a (cos.f64 (*.f64 (*.f64 1/180 (PI.f64)) angle)))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/180 (*.f64 (PI.f64) angle))))) (*.f64 a (cos.f64 (*.f64 (*.f64 1/180 (PI.f64)) angle)))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 13 points increase in error, 10 points decrease in error
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (*.f64 1/180 (Rewrite<= *-commutative_binary64 (*.f64 angle (PI.f64)))))) (*.f64 a (cos.f64 (*.f64 (*.f64 1/180 (PI.f64)) angle)))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/180 (*.f64 (PI.f64) angle)))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 7 points increase in error, 8 points decrease in error
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (Rewrite<= *-commutative_binary64 (*.f64 angle (PI.f64))))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 71 points increase in error, 13 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 b b) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))) (*.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 17 points increase in error, 6 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 a a) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 3 points increase in error, 7 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))))) (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) -1/4)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)))) (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8)))) -1/4)): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)))))) -1/4): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -1/4 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))))))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in angle around 0 35.3

      \[\leadsto \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a \cdot \color{blue}{1}\right) \cdot \left(\left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot -0.25\right) \]

   7. Applied egg-rr 35.1

      \[\leadsto \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a \cdot 1\right) \cdot \left(\color{blue}{\left(0 + x-scale \cdot {16}^{0.5}\right)} \cdot -0.25\right) \]

   if -1.73717618817821823e-59 < x-scale < 2.7302332699994509e-27

   1. Initial program 63.6

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

   2. Taylor expanded in angle around 0 50.8

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

   3. Applied egg-rr 52.0

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

   4. Applied egg-rr 41.6

      \[\leadsto 0.25 \cdot e^{\log \color{blue}{\left(\sqrt{{\left(\sqrt[3]{\left(b \cdot y-scale\right) \cdot 4}\right)}^{2}}\right)} \cdot 3} \]

   if 2.7302332699994509e-27 < x-scale

   1. Initial program 63.5

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

   2. Taylor expanded in x-scale around inf 59.8

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

   3. Simplified 59.0

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale} \cdot \frac{a \cdot a}{y-scale} + \frac{b \cdot b}{y-scale} \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale}\right)}} \]
      Proof
      (*.f64 (*.f64 1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (*.f64 a a) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 x-scale y-scale) (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (*.f64 a a) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 18 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 y-scale x-scale)) (sqrt.f64 8))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (*.f64 a a) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (Rewrite<= associate-*r*_binary64 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8))))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (*.f64 a a) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 11 points increase in error, 15 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 a 2)) (*.f64 y-scale y-scale))) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 22 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.f64 y-scale y-scale)) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2))) (*.f64 (/.f64 (*.f64 b b) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2)) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) y-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) y-scale)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 y-scale y-scale))))))): 2 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2)) (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2))) (*.f64 2 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 1/4 (*.f64 (*.f64 y-scale (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 2 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2))) (*.f64 2 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 y-scale 2)))))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in y-scale around 0 48.7

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

   5. Simplified 40.5

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a \cdot \cos \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{\sqrt{2}}{y-scale}\right)} \]
      Proof
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (*.f64 (*.f64 1/180 (PI.f64)) angle))) (*.f64 a (cos.f64 (*.f64 (*.f64 1/180 (PI.f64)) angle)))) (/.f64 (sqrt.f64 2) y-scale)): 0 points increase in error, 0 points decrease in error
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/180 (*.f64 (PI.f64) angle))))) (*.f64 a (cos.f64 (*.f64 (*.f64 1/180 (PI.f64)) angle)))) (/.f64 (sqrt.f64 2) y-scale)): 13 points increase in error, 8 points decrease in error
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (*.f64 1/180 (Rewrite<= *-commutative_binary64 (*.f64 angle (PI.f64)))))) (*.f64 a (cos.f64 (*.f64 (*.f64 1/180 (PI.f64)) angle)))) (/.f64 (sqrt.f64 2) y-scale)): 0 points increase in error, 0 points decrease in error
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/180 (*.f64 (PI.f64) angle)))))) (/.f64 (sqrt.f64 2) y-scale)): 6 points increase in error, 6 points decrease in error
      (*.f64 (hypot.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (Rewrite<= *-commutative_binary64 (*.f64 angle (PI.f64))))))) (/.f64 (sqrt.f64 2) y-scale)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) (/.f64 (sqrt.f64 2) y-scale)): 38 points increase in error, 17 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 b b) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))) (*.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (/.f64 (sqrt.f64 2) y-scale)): 15 points increase in error, 5 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (/.f64 (sqrt.f64 2) y-scale)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (/.f64 (sqrt.f64 2) y-scale)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 a a) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) (/.f64 (sqrt.f64 2) y-scale)): 5 points increase in error, 6 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (/.f64 (sqrt.f64 2) y-scale)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))))) (/.f64 (sqrt.f64 2) y-scale)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (sqrt.f64 2) y-scale) (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)))))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in x-scale around 0 45.1

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

   7. Simplified 34.0

      \[\leadsto \color{blue}{\mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \sqrt{8}\right)\right)} \]
      Proof
      (*.f64 (hypot.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 (*.f64 1/4 (sqrt.f64 2)) (*.f64 x-scale (sqrt.f64 8)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) (*.f64 (*.f64 1/4 (sqrt.f64 2)) (*.f64 x-scale (sqrt.f64 8)))): 72 points increase in error, 12 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 a a) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))) (*.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (*.f64 (*.f64 1/4 (sqrt.f64 2)) (*.f64 x-scale (sqrt.f64 8)))): 5 points increase in error, 5 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (*.f64 (*.f64 1/4 (sqrt.f64 2)) (*.f64 x-scale (sqrt.f64 8)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.f64 (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (*.f64 (*.f64 1/4 (sqrt.f64 2)) (*.f64 x-scale (sqrt.f64 8)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 b b) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) (*.f64 (*.f64 1/4 (sqrt.f64 2)) (*.f64 x-scale (sqrt.f64 8)))): 19 points increase in error, 4 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) (*.f64 (*.f64 1/4 (sqrt.f64 2)) (*.f64 x-scale (sqrt.f64 8)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))))) (*.f64 (*.f64 1/4 (sqrt.f64 2)) (*.f64 x-scale (sqrt.f64 8)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)))) (Rewrite<= associate-*r*_binary64 (*.f64 1/4 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/4 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8)))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 1/4 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 b 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))))))): 1 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 38.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.7371761881782182 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right) \cdot \left(\left(x-scale \cdot {16}^{0.5}\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq 2.730233269999451 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot e^{\log \left(\sqrt{{\left(\sqrt[3]{4 \cdot \left(b \cdot y-scale\right)}\right)}^{2}}\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	40.2
Cost	33480

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right)\\ \mathbf{if}\;x-scale \leq -1.7371761881782182 \cdot 10^{-59}:\\ \;\;\;\;t_0 \cdot \left(\left(x-scale \cdot {16}^{0.5}\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;0.25 \cdot e^{\log \left(\sqrt{{\left(\sqrt[3]{4 \cdot \left(b \cdot y-scale\right)}\right)}^{2}}\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \left(t_0 \cdot \frac{\sqrt{2}}{y-scale}\right)\\ \end{array} \]

Alternative 2
Error	41.9
Cost	32904

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -1.7371761881782182 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right) \cdot \left(\left(x-scale \cdot {16}^{0.5}\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq 5.5 \cdot 10^{+71}:\\ \;\;\;\;0.25 \cdot e^{\log \left(\sqrt{{\left(\sqrt[3]{4 \cdot \left(b \cdot y-scale\right)}\right)}^{2}}\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \left(-0.25 \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 3
Error	46.7
Cost	26760

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -1.7371761881782182 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{hypot}\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right), a\right) \cdot \left(-0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;x-scale \leq 5.2 \cdot 10^{+62}:\\ \;\;\;\;0.25 \cdot \sqrt{{\left(4 \cdot \left(b \cdot y-scale\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \left(-0.25 \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 4
Error	45.8
Cost	26760

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -2.6005655341415004 \cdot 10^{-251}:\\ \;\;\;\;\mathsf{hypot}\left(b \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right), a\right) \cdot \left(\left(x-scale \cdot {16}^{0.5}\right) \cdot -0.25\right)\\ \mathbf{elif}\;x-scale \leq 5.2 \cdot 10^{+62}:\\ \;\;\;\;0.25 \cdot \sqrt{{\left(4 \cdot \left(b \cdot y-scale\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \left(-0.25 \cdot \sqrt{2}\right)\\ \end{array} \]

Alternative 5
Error	51.3
Cost	14168

\[\begin{array}{l} t_0 := a \cdot \sqrt{8}\\ t_1 := t_0 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot -0.25\right)\right)\\ t_2 := 0.25 \cdot \sqrt{{\left(4 \cdot \left(b \cdot y-scale\right)\right)}^{2}}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{+110}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;b \leq -1.2950944798119793 \cdot 10^{-61}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)}\right)\\ \mathbf{elif}\;b \leq -9.0142365821779 \cdot 10^{-89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 8.702122679757685 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 5.102836683145532 \cdot 10^{-214}:\\ \;\;\;\;0.25 \cdot \left(t_0 \cdot \left(x-scale \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;b \leq 4.218529701371672 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	51.3
Cost	14168

\[\begin{array}{l} t_0 := a \cdot \sqrt{8}\\ t_1 := t_0 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot -0.25\right)\right)\\ t_2 := 0.25 \cdot \sqrt{{\left(4 \cdot \left(b \cdot y-scale\right)\right)}^{2}}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{+110}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;b \leq -1.2950944798119793 \cdot 10^{-61}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)}\right)\\ \mathbf{elif}\;b \leq -9.0142365821779 \cdot 10^{-89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 8.702122679757685 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 5.102836683145532 \cdot 10^{-214}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot t_0\right)\right)\\ \mathbf{elif}\;b \leq 4.218529701371672 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	51.3
Cost	13904

\[\begin{array}{l} t_0 := 0.25 \cdot \sqrt{{\left(4 \cdot \left(b \cdot y-scale\right)\right)}^{2}}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{+110}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{elif}\;b \leq -1.2950944798119793 \cdot 10^{-61}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)}\right)\\ \mathbf{elif}\;b \leq -9.0142365821779 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.218529701371672 \cdot 10^{-135}:\\ \;\;\;\;\left(a \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	52.2
Cost	13444

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 3.958234779783596 \cdot 10^{-207}:\\ \;\;\;\;0.25 \cdot \sqrt{{\left(4 \cdot \left(b \cdot y-scale\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)}\right)\\ \end{array} \]

Alternative 9
Error	53.0
Cost	7236

\[\begin{array}{l} \mathbf{if}\;y-scale \leq 5.432808380506408 \cdot 10^{-223}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)}\right)\\ \end{array} \]

Alternative 10
Error	54.0
Cost	192

\[b \cdot y-scale \]

Reproduce

herbie shell --seed 2022312 
(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))))