b from scale-rotated-ellipse

Average Accuracy: 0.1% → 41.5%

Time: 2.4min

Precision: binary64

Cost: 19912

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-113}:\\ \;\;\;\;0.25 \cdot \left|\left(a \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot x-scale\right)\right|\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+126}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{0.125} \cdot \left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right)\right|\\ \end{array} \]

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
 (if (<= b -2.3e-113)
   (* 0.25 (fabs (* (* a (sqrt 8.0)) (* (sqrt 2.0) x-scale))))
   (if (<= b 2e+126)
     0.0
     (fabs (* (sqrt 0.125) (* (sqrt 8.0) (* a x-scale)))))))

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 tmp;
	if (b <= -2.3e-113) {
		tmp = 0.25 * fabs(((a * sqrt(8.0)) * (sqrt(2.0) * x_45_scale)));
	} else if (b <= 2e+126) {
		tmp = 0.0;
	} else {
		tmp = fabs((sqrt(0.125) * (sqrt(8.0) * (a * x_45_scale))));
	}
	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 tmp;
	if (b <= -2.3e-113) {
		tmp = 0.25 * Math.abs(((a * Math.sqrt(8.0)) * (Math.sqrt(2.0) * x_45_scale)));
	} else if (b <= 2e+126) {
		tmp = 0.0;
	} else {
		tmp = Math.abs((Math.sqrt(0.125) * (Math.sqrt(8.0) * (a * x_45_scale))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, 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))

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= -2.3e-113:
		tmp = 0.25 * math.fabs(((a * math.sqrt(8.0)) * (math.sqrt(2.0) * x_45_scale)))
	elif b <= 2e+126:
		tmp = 0.0
	else:
		tmp = math.fabs((math.sqrt(0.125) * (math.sqrt(8.0) * (a * x_45_scale))))
	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)
	tmp = 0.0
	if (b <= -2.3e-113)
		tmp = Float64(0.25 * abs(Float64(Float64(a * sqrt(8.0)) * Float64(sqrt(2.0) * x_45_scale))));
	elseif (b <= 2e+126)
		tmp = 0.0;
	else
		tmp = abs(Float64(sqrt(0.125) * Float64(sqrt(8.0) * Float64(a * x_45_scale))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= -2.3e-113)
		tmp = 0.25 * abs(((a * sqrt(8.0)) * (sqrt(2.0) * x_45_scale)));
	elseif (b <= 2e+126)
		tmp = 0.0;
	else
		tmp = abs((sqrt(0.125) * (sqrt(8.0) * (a * x_45_scale))));
	end
	tmp_2 = 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_] := If[LessEqual[b, -2.3e-113], N[(0.25 * N[Abs[N[(N[(a * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+126], 0.0, N[Abs[N[(N[Sqrt[0.125], $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(a * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.30000000000000008e-113

   1. Initial program 0.1%

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

   2. Simplified 0.2%

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

      [Start] 0.1

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

   3. Taylor expanded in angle around 0 25.4%

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

   4. Applied egg-rr 37.8%

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

      [Start] 25.4	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      add-sqr-sqrt [=>] 23.9	\[ 0.25 \cdot \color{blue}{\left(\sqrt{\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)} \cdot \sqrt{\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)}\right)} \]
      sqrt-unprod [=>] 30.5	\[ 0.25 \cdot \color{blue}{\sqrt{\left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}} \]
      rem-sqrt-square [=>] 37.8	\[ 0.25 \cdot \color{blue}{\left|\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right|} \]
      associate-*r* [=>] 37.8	\[ 0.25 \cdot \left|\color{blue}{\left(\sqrt{2} \cdot x-scale\right) \cdot \left(a \cdot \sqrt{8}\right)}\right| \]
      *-commutative [=>] 37.8	\[ 0.25 \cdot \left|\color{blue}{\left(a \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot x-scale\right)}\right| \]

   if -2.30000000000000008e-113 < b < 1.99999999999999985e126

   1. Initial program 0.1%

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

   2. Simplified 0.2%

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

      [Start] 0.1

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

   3. Taylor expanded in angle around 0 19.6%

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

   4. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {\left(a \cdot x-scale\right)}^{2}\right) \cdot 0.125}} \]
      Proof

      [Start] 19.6	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      add-sqr-sqrt [=>] 18.1	\[ \color{blue}{\sqrt{0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)} \cdot \sqrt{0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}} \]
      sqrt-unprod [=>] 25.9	\[ \color{blue}{\sqrt{\left(0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\right) \cdot \left(0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\right)}} \]
      associate-*r* [=>] 25.9	\[ \sqrt{\left(0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\right) \cdot \color{blue}{\left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}} \]
      associate-*r* [=>] 25.9	\[ \sqrt{\color{blue}{\left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)} \cdot \left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)} \]
      *-commutative [=>] 25.9	\[ \sqrt{\color{blue}{\left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(0.25 \cdot \sqrt{2}\right)\right)} \cdot \left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)} \]
      *-commutative [=>] 25.9	\[ \sqrt{\left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(0.25 \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(0.25 \cdot \sqrt{2}\right)\right)}} \]
      swap-sqr [=>] 25.9	\[ \sqrt{\color{blue}{\left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(0.25 \cdot \sqrt{2}\right)\right)}} \]

   5. Simplified 26.0%

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

      [Start] 26.0	\[ \sqrt{\left(8 \cdot {\left(a \cdot x-scale\right)}^{2}\right) \cdot 0.125} \]
      *-commutative [=>] 26.0	\[ \sqrt{\color{blue}{0.125 \cdot \left(8 \cdot {\left(a \cdot x-scale\right)}^{2}\right)}} \]
      associate-*r* [=>] 26.0	\[ \sqrt{\color{blue}{\left(0.125 \cdot 8\right) \cdot {\left(a \cdot x-scale\right)}^{2}}} \]
      metadata-eval [=>] 26.0	\[ \sqrt{\color{blue}{1} \cdot {\left(a \cdot x-scale\right)}^{2}} \]
      *-lft-identity [=>] 26.0	\[ \sqrt{\color{blue}{{\left(a \cdot x-scale\right)}^{2}}} \]
      *-commutative [=>] 26.0	\[ \sqrt{{\color{blue}{\left(x-scale \cdot a\right)}}^{2}} \]

   6. Applied egg-rr 29.3%

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

      [Start] 26.0	\[ \sqrt{{\left(x-scale \cdot a\right)}^{2}} \]
      sqrt-pow1 [=>] 19.7	\[ \color{blue}{{\left(x-scale \cdot a\right)}^{\left(\frac{2}{2}\right)}} \]
      metadata-eval [=>] 19.7	\[ {\left(x-scale \cdot a\right)}^{\color{blue}{1}} \]
      pow1 [<=] 19.7	\[ \color{blue}{x-scale \cdot a} \]
      expm1-log1p-u [=>] 19.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x-scale \cdot a\right)\right)} \]
      expm1-udef [=>] 28.9	\[ \color{blue}{e^{\mathsf{log1p}\left(x-scale \cdot a\right)} - 1} \]
      log1p-udef [=>] 28.9	\[ e^{\color{blue}{\log \left(1 + x-scale \cdot a\right)}} - 1 \]
      add-exp-log [<=] 29.3	\[ \color{blue}{\left(1 + x-scale \cdot a\right)} - 1 \]

   7. Taylor expanded in x-scale around 0 43.3%

      \[\leadsto \color{blue}{1} - 1 \]

   if 1.99999999999999985e126 < b

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0

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

   3. Taylor expanded in angle around 0 30.0%

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

   4. Applied egg-rr 43.7%

      \[\leadsto \color{blue}{\left|\sqrt{0.125} \cdot \left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right)\right|} \]
      Proof

      [Start] 30.0	\[ 0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \]
      add-sqr-sqrt [=>] 28.7	\[ \color{blue}{\sqrt{0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)} \cdot \sqrt{0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}} \]
      sqr-abs [<=] 28.7	\[ \color{blue}{\left|\sqrt{0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right| \cdot \left|\sqrt{0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right|} \]
      mul-fabs [=>] 28.7	\[ \color{blue}{\left|\sqrt{0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)} \cdot \sqrt{0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right|} \]
      sqrt-unprod [=>] 33.5	\[ \left|\color{blue}{\sqrt{\left(0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\right) \cdot \left(0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\right)}}\right| \]
      associate-*r* [=>] 33.5	\[ \left|\sqrt{\color{blue}{\left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)} \cdot \left(0.25 \cdot \left(\sqrt{2} \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)\right)}\right| \]
      associate-*r* [=>] 33.5	\[ \left|\sqrt{\left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right) \cdot \color{blue}{\left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}}\right| \]
      swap-sqr [=>] 33.4	\[ \left|\sqrt{\color{blue}{\left(\left(0.25 \cdot \sqrt{2}\right) \cdot \left(0.25 \cdot \sqrt{2}\right)\right) \cdot \left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}}\right| \]
      swap-sqr [=>] 33.4	\[ \left|\sqrt{\color{blue}{\left(\left(0.25 \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right| \]
      metadata-eval [=>] 33.4	\[ \left|\sqrt{\left(\color{blue}{0.0625} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right) \cdot \left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right| \]
      add-sqr-sqrt [<=] 33.5	\[ \left|\sqrt{\left(0.0625 \cdot \color{blue}{2}\right) \cdot \left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right| \]
      metadata-eval [=>] 33.5	\[ \left|\sqrt{\color{blue}{0.125} \cdot \left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right| \]
      metadata-eval [<=] 33.5	\[ \left|\sqrt{\color{blue}{\sqrt{0.015625}} \cdot \left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right| \]
      metadata-eval [<=] 33.5	\[ \left|\sqrt{\sqrt{\color{blue}{0.0625 \cdot 0.25}} \cdot \left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right| \]
      metadata-eval [<=] 33.5	\[ \left|\sqrt{\sqrt{\color{blue}{\left(0.25 \cdot 0.25\right)} \cdot 0.25} \cdot \left(\left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right) \cdot \left(x-scale \cdot \left(a \cdot \sqrt{8}\right)\right)\right)}\right| \]

3. Recombined 3 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-113}:\\ \;\;\;\;0.25 \cdot \left|\left(a \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot x-scale\right)\right|\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+126}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{0.125} \cdot \left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right)\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	41.5%
Cost	19913

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-113} \lor \neg \left(b \leq 7.2 \cdot 10^{+126}\right):\\ \;\;\;\;\left|\sqrt{0.125} \cdot \left(\sqrt{8} \cdot \left(a \cdot x-scale\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Accuracy	41.4%
Cost	19784

\[\begin{array}{l} t_0 := \sqrt[3]{a \cdot x-scale}\\ \mathbf{if}\;b \leq -2.3 \cdot 10^{-113}:\\ \;\;\;\;{\left(\left|t_0\right|\right)}^{3}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+126}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;{\left({t_0}^{2}\right)}^{1.5}\\ \end{array} \]

Alternative 3
Accuracy	41.4%
Cost	19721

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-113} \lor \neg \left(b \leq 2 \cdot 10^{+126}\right):\\ \;\;\;\;{\left(\left|\sqrt[3]{a \cdot x-scale}\right|\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Accuracy	36.4%
Cost	13188

\[\begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{{\left(a \cdot x-scale\right)}^{2}}\\ \mathbf{elif}\;b \leq 10^{+128}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]

Alternative 5
Accuracy	36.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{x-scale \cdot \left(a \cdot \left(a \cdot x-scale\right)\right)}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+127}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]

Alternative 6
Accuracy	35.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+202}:\\ \;\;\;\;a \cdot x-scale\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+126}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;a \cdot x-scale\\ \end{array} \]

Alternative 7
Accuracy	23.2%
Cost	192

\[a \cdot x-scale \]

Reproduce

herbie shell --seed 2023137 
(FPCore (a b angle x-scale y-scale)
  :name "b from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (- (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))