a from scale-rotated-ellipse

Average Error: 63.5 → 41.9

Time: 1.8min

Precision: binary64

Cost: 59464

Math

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

↓

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := a \cdot \sin t_0\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;y-scale \leq -1.4188156489090483 \cdot 10^{-208}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(y-scale \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{hypot}\left(t_1, b \cdot \cos t_0\right) \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 8.112591889012834 \cdot 10^{-120}:\\ \;\;\;\;\left(0.25 \cdot \left(\sqrt{8} \cdot x-scale\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos t_2}^{2}, {\sin t_2}^{2} \cdot \left(b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(b, t_1\right)\right)\right)\\ \end{array} \]

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 (* angle (* PI 0.005555555555555556)))
        (t_1 (* a (sin t_0)))
        (t_2 (* 0.005555555555555556 (* angle PI))))
   (if (<= y-scale -1.4188156489090483e-208)
     (*
      (* (sqrt 8.0) (* y-scale (sqrt 2.0)))
      (* (hypot t_1 (* b (cos t_0))) -0.25))
     (if (<= y-scale 8.112591889012834e-120)
       (*
        (* 0.25 (* (sqrt 8.0) x-scale))
        (sqrt
         (*
          2.0
          (fma (* a a) (pow (cos t_2) 2.0) (* (pow (sin t_2) 2.0) (* b b))))))
       (* (sqrt 2.0) (* 0.25 (* (* y-scale (sqrt 8.0)) (hypot b t_1))))))))

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 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_1 = a * sin(t_0);
	double t_2 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (y_45_scale <= -1.4188156489090483e-208) {
		tmp = (sqrt(8.0) * (y_45_scale * sqrt(2.0))) * (hypot(t_1, (b * cos(t_0))) * -0.25);
	} else if (y_45_scale <= 8.112591889012834e-120) {
		tmp = (0.25 * (sqrt(8.0) * x_45_scale)) * sqrt((2.0 * fma((a * a), pow(cos(t_2), 2.0), (pow(sin(t_2), 2.0) * (b * b)))));
	} else {
		tmp = sqrt(2.0) * (0.25 * ((y_45_scale * sqrt(8.0)) * hypot(b, t_1)));
	}
	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(angle * Float64(pi * 0.005555555555555556))
	t_1 = Float64(a * sin(t_0))
	t_2 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (y_45_scale <= -1.4188156489090483e-208)
		tmp = Float64(Float64(sqrt(8.0) * Float64(y_45_scale * sqrt(2.0))) * Float64(hypot(t_1, Float64(b * cos(t_0))) * -0.25));
	elseif (y_45_scale <= 8.112591889012834e-120)
		tmp = Float64(Float64(0.25 * Float64(sqrt(8.0) * x_45_scale)) * sqrt(Float64(2.0 * fma(Float64(a * a), (cos(t_2) ^ 2.0), Float64((sin(t_2) ^ 2.0) * Float64(b * b))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(0.25 * Float64(Float64(y_45_scale * sqrt(8.0)) * hypot(b, t_1))));
	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[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -1.4188156489090483e-208], N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[(y$45$scale * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$1 ^ 2 + N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 8.112591889012834e-120], N[(N[(0.25 * N[(N[Sqrt[8.0], $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(a * a), $MachinePrecision] * N[Power[N[Cos[t$95$2], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 * N[(N[(y$45$scale * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[b ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < -1.4188156489090483e-208

   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 y-scale around -inf 60.4

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

   3. Simplified 59.0

      \[\leadsto \color{blue}{\left(-0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \left(\frac{a \cdot a}{x-scale} \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{\frac{x-scale}{b} \cdot \frac{x-scale}{b}}\right)}} \]
      Proof
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (*.f64 a a) x-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) x-scale)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (/.f64 x-scale b) (/.f64 x-scale b))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) x-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) x-scale)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (/.f64 x-scale b) (/.f64 x-scale b))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 x-scale x-scale))) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (/.f64 x-scale b) (/.f64 x-scale b))))))): 19 points increase in error, 6 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2))) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (/.f64 x-scale b) (/.f64 x-scale b))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x-scale x-scale) (*.f64 b b)))))))): 14 points increase in error, 9 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2)) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (/.f64 (pow.f64 x-scale 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 b 2)) (pow.f64 x-scale 2))))))): 3 points increase in error, 1 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (pow.f64 x-scale 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2))) (*.f64 2 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1/4 (*.f64 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 2 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2))) (*.f64 2 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)))))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in x-scale around -inf 62.6

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

   5. Simplified 62.6

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot b, a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.25\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))) b) (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/180))) b) (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))))) 1/4)): 11 points increase in error, 7 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/180 (*.f64 angle (PI.f64))))) b) (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 a (sin.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/180)))))) 1/4)): 17 points increase in error, 8 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 a (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b)) (*.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))))) 1/4)): 73 points increase in error, 6 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b b))) (*.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) 1/4)): 4 points increase in error, 4 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 b b)) (*.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2))) (*.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 b 2)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 a a) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))))) 1/4)): 20 points increase in error, 2 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 b 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 b 2)) (*.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.f64 (pow.f64 a 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<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 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
      (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)))))) 1/4): 31 points increase in error, 24 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/4 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 47.2

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

   7. Taylor expanded in y-scale around -inf 49.0

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

   8. Simplified 39.3

      \[\leadsto \color{blue}{\left(\sqrt{8} \cdot \left(y-scale \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot -0.25\right)} \]
      Proof
      (*.f64 (*.f64 (sqrt.f64 8) (*.f64 y-scale (sqrt.f64 2))) (*.f64 (hypot.f64 (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))) (*.f64 b (cos.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 8) y-scale) (sqrt.f64 2))) (*.f64 (hypot.f64 (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))) (*.f64 b (cos.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 y-scale (sqrt.f64 8))) (sqrt.f64 2)) (*.f64 (hypot.f64 (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))) (*.f64 b (cos.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8)))) (*.f64 (hypot.f64 (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))) (*.f64 b (cos.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (hypot.f64 (*.f64 a (sin.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/180)))) (*.f64 b (cos.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))))) -1/4)): 14 points increase in error, 14 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (hypot.f64 (*.f64 a (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 b (cos.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (hypot.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b (cos.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/180))))) -1/4)): 8 points increase in error, 9 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (hypot.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/180 (*.f64 angle (PI.f64))))))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (hypot.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b))))) -1/4)): 68 points increase in error, 11 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (sqrt.f64 (+.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 a a) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))) (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b)))) -1/4)): 17 points increase in error, 6 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (sqrt.f64 (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b)))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b)))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b b))))) -1/4)): 3 points increase in error, 4 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (Rewrite<= unpow2_binary64 (pow.f64 b 2))))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (sqrt.f64 (+.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 b 2)))) -1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 b 2)) (*.f64 (pow.f64 a 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
      (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (*.f64 (sqrt.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.f64 (pow.f64 a 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<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 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 y-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))))))): 0 points increase in error, 0 points decrease in error

   if -1.4188156489090483e-208 < y-scale < 8.1125918890128341e-120

   1. Initial program 63.9

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

   2. Taylor expanded in y-scale around 0 51.8

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

   3. Simplified 51.8

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]
      Proof
      (*.f64 (*.f64 1/4 (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (*.f64 2 (fma.f64 (*.f64 a a) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (*.f64 2 (fma.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (*.f64 2 (fma.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (*.f64 2 (fma.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (Rewrite<= *-commutative_binary64 (*.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
      (*.f64 (*.f64 1/4 (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (*.f64 2 (Rewrite<= fma-def_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))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/4 (*.f64 x-scale (sqrt.f64 8))) (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.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 x-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 2 (*.f64 (pow.f64 a 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.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

   if 8.1125918890128341e-120 < y-scale

   1. Initial program 63.3

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

   2. Taylor expanded in y-scale around -inf 59.8

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

   3. Simplified 57.9

      \[\leadsto \color{blue}{\left(-0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{2 \cdot \left(\frac{a \cdot a}{x-scale} \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale} + \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{\frac{x-scale}{b} \cdot \frac{x-scale}{b}}\right)}} \]
      Proof
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (*.f64 a a) x-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) x-scale)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (/.f64 x-scale b) (/.f64 x-scale b))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) x-scale) (/.f64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) x-scale)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (/.f64 x-scale b) (/.f64 x-scale b))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 x-scale x-scale))) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (/.f64 x-scale b) (/.f64 x-scale b))))))): 19 points increase in error, 6 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2))) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (*.f64 (/.f64 x-scale b) (/.f64 x-scale b))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x-scale x-scale) (*.f64 b b)))))))): 14 points increase in error, 9 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2)) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (/.f64 (pow.f64 x-scale 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 b 2)) (pow.f64 x-scale 2))))))): 3 points increase in error, 1 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (pow.f64 x-scale 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)) (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/4 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8)))) (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2))) (*.f64 2 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1/4 (*.f64 (*.f64 x-scale (*.f64 y-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 2 (/.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2))) (*.f64 2 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (pow.f64 x-scale 2)))))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in x-scale around -inf 47.4

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

   5. Simplified 36.5

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot b, a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.25\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))) b) (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/180))) b) (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))))) 1/4)): 11 points increase in error, 7 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/180 (*.f64 angle (PI.f64))))) b) (*.f64 a (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180)))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 a (sin.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/180)))))) 1/4)): 17 points increase in error, 8 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (hypot.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 a (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) b)) (*.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))))) 1/4)): 73 points increase in error, 6 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 b b))) (*.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) 1/4)): 4 points increase in error, 4 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 b b)) (*.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2))) (*.f64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 b 2)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 a a) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))))) 1/4)): 20 points increase in error, 2 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 b 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2) (pow.f64 b 2)) (*.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)))))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2))) (*.f64 (pow.f64 a 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<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 (*.f64 y-scale (sqrt.f64 8)) (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 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
      (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 2) (pow.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)))))) 1/4): 31 points increase in error, 24 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/4 (*.f64 (*.f64 (sqrt.f64 2) (*.f64 y-scale (sqrt.f64 8))) (sqrt.f64 (+.f64 (*.f64 (pow.f64 b 2) (pow.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) 2)) (*.f64 (pow.f64 a 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 36.5

      \[\leadsto \sqrt{2} \cdot \left(\left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\color{blue}{1} \cdot b, a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot 0.25\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -1.4188156489090483 \cdot 10^{-208}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(y-scale \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot -0.25\right)\\ \mathbf{elif}\;y-scale \leq 8.112591889012834 \cdot 10^{-120}:\\ \;\;\;\;\left(0.25 \cdot \left(\sqrt{8} \cdot x-scale\right)\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(a \cdot a, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(b, a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	41.8
Cost	46212

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := a \cdot \sin t_0\\ \mathbf{if}\;y-scale \leq -3.35975689536478 \cdot 10^{-306}:\\ \;\;\;\;\left(\sqrt{8} \cdot \left(y-scale \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{hypot}\left(t_1, b \cdot \cos t_0\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(b, t_1\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	42.1
Cost	33488

\[\begin{array}{l} t_0 := \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ t_1 := \mathsf{hypot}\left(b, a \cdot t_0\right)\\ \mathbf{if}\;y-scale \leq -9 \cdot 10^{+150}:\\ \;\;\;\;0.25 \cdot e^{\log \left(\sqrt{{\left(\sqrt[3]{4 \cdot \left(y-scale \cdot b\right)}\right)}^{2}}\right) \cdot 3}\\ \mathbf{elif}\;y-scale \leq -2.4694390721855824 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(t_1 \cdot \sqrt{8 \cdot \left(y-scale \cdot y-scale\right)}\right)\right)\\ \mathbf{elif}\;y-scale \leq -5.32618222859431 \cdot 10^{-189}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\left(y-scale \cdot \left(\sqrt{8} \cdot b\right)\right) \cdot \left(t_0 \cdot \frac{\sqrt{2}}{y-scale}\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.2080094369650388 \cdot 10^{-194}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot t_1\right)\right)\\ \end{array} \]

Alternative 3
Error	43.7
Cost	33224

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -4.531227277493079 \cdot 10^{-76}:\\ \;\;\;\;0.25 \cdot e^{\log \left(\sqrt{{\left(\sqrt[3]{4 \cdot \left(y-scale \cdot b\right)}\right)}^{2}}\right) \cdot 3}\\ \mathbf{elif}\;y-scale \leq 2.2080094369650388 \cdot 10^{-194}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(b, a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	45.9
Cost	32904

\[\begin{array}{l} t_0 := 0.25 \cdot e^{\log \left(\sqrt{{\left(\sqrt[3]{4 \cdot \left(y-scale \cdot b\right)}\right)}^{2}}\right) \cdot 3}\\ \mathbf{if}\;b \leq -4.565078898764429 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.531360431138719 \cdot 10^{-164}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	51.1
Cost	26628

\[\begin{array}{l} \mathbf{if}\;b \leq -1.0813573827787665 \cdot 10^{-93}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(b \cdot \left(\sqrt{8} \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.531360431138719 \cdot 10^{-164}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+168}:\\ \;\;\;\;\sqrt{{\left(y-scale \cdot b\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 6
Error	51.1
Cost	13640

\[\begin{array}{l} \mathbf{if}\;b \leq -1.0813573827787665 \cdot 10^{-93}:\\ \;\;\;\;-0.25 \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.531360431138719 \cdot 10^{-164}:\\ \;\;\;\;-0.25 \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \left(\sqrt{8} \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+168}:\\ \;\;\;\;\sqrt{{\left(y-scale \cdot b\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 7
Error	53.3
Cost	13320

\[\begin{array}{l} t_0 := \sqrt{{\left(y-scale \cdot b\right)}^{2}}\\ \mathbf{if}\;angle \leq -3.3151914762656406 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 2.1723894497818595 \cdot 10^{-182}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	53.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{+121}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{elif}\;b \leq -8.933900570257289 \cdot 10^{-62}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \sqrt{2 \cdot \left(8 \cdot \left(b \cdot b\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]

Alternative 9
Error	53.7
Cost	192

\[y-scale \cdot b \]

Reproduce

herbie shell --seed 2022294 
(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))))