Simplification of discriminant from scale-rotated-ellipse

Average Error: 41.1 → 5.7

Time: 1.3min

Precision: binary64

Cost: 1220

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x-scale \leq -7.5 \cdot 10^{-60}:\\ \;\;\;\;-4 \cdot \frac{\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale -7.5e-60)
   (* -4.0 (/ (/ (/ a x-scale) (/ y-scale b)) (* (/ y-scale b) (/ x-scale a))))
   (*
    -4.0
    (/ (* (/ a y-scale) (/ b x-scale)) (/ (/ x-scale b) (/ a y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((((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 * (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)) - ((4.0 * (((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));
}

↓

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= -7.5e-60) {
		tmp = -4.0 * (((a / x_45_scale) / (y_45_scale / b)) / ((y_45_scale / b) * (x_45_scale / a)));
	} else {
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((x_45_scale / b) / (a / y_45_scale)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((((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 * (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)) - ((4.0 * (((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));
}

↓

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= -7.5e-60) {
		tmp = -4.0 * (((a / x_45_scale) / (y_45_scale / b)) / ((y_45_scale / b) * (x_45_scale / a)));
	} else {
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((x_45_scale / b) / (a / y_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((((((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 * (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)) - ((4.0 * (((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))

↓

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= -7.5e-60:
		tmp = -4.0 * (((a / x_45_scale) / (y_45_scale / b)) / ((y_45_scale / b) * (x_45_scale / a)))
	else:
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((x_45_scale / b) / (a / y_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(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) * 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)) - Float64(Float64(4.0 * 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)))
end

↓

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= -7.5e-60)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / x_45_scale) / Float64(y_45_scale / b)) / Float64(Float64(y_45_scale / b) * Float64(x_45_scale / a))));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale)) / Float64(Float64(x_45_scale / b) / Float64(a / y_45_scale))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= -7.5e-60)
		tmp = -4.0 * (((a / x_45_scale) / (y_45_scale / b)) / ((y_45_scale / b) * (x_45_scale / a)));
	else
		tmp = -4.0 * (((a / y_45_scale) * (b / x_45_scale)) / ((x_45_scale / b) / (a / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(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] * 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]), $MachinePrecision] - N[(N[(4.0 * 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]), $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]), $MachinePrecision]

↓

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, -7.5e-60], N[(-4.0 * N[(N[(N[(a / x$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale / b), $MachinePrecision] * N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale / b), $MachinePrecision] / N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < -7.5000000000000002e-60

   1. Initial program 38.7

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

   2. Simplified 43.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale}, \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale} \cdot -4, \frac{4 \cdot \left(\left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)\right)}{x-scale \cdot \left(y-scale \cdot \frac{x-scale}{\cos \left(\frac{angle}{180} \cdot \pi\right)}\right)} \cdot \frac{\cos \left(\frac{angle}{180} \cdot \pi\right)}{y-scale}\right)} \]
      Proof
      (fma.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) (*.f64 x-scale x-scale)) (*.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) (*.f64 y-scale y-scale)) -4) (*.f64 (/.f64 (*.f64 4 (*.f64 (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale)) (*.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) (*.f64 y-scale y-scale)) -4) (*.f64 (/.f64 (*.f64 4 (*.f64 (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 31 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) -4) (*.f64 (/.f64 (*.f64 4 (*.f64 (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 31 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale) (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 (/.f64 (*.f64 4 (*.f64 (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 0 points increase in error, 31 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale) 4))) (*.f64 (/.f64 (*.f64 4 (*.f64 (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 0 points increase in error, 31 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4)) (*.f64 (/.f64 (*.f64 4 (*.f64 (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 31 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 2 2)) (*.f64 (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 0 points increase in error, 31 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (*.f64 (*.f64 2 2) (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 0 points increase in error, 2 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (*.f64 (*.f64 2 2) (*.f64 (*.f64 (-.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 20 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (*.f64 (*.f64 2 2) (*.f64 (*.f64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 a a)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 9 points increase in error, 6 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (*.f64 (*.f64 2 2) (*.f64 (*.f64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (-.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 17 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) (*.f64 2 (*.f64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 0 points increase in error, 10 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) (*.f64 2 (*.f64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 10 points increase in error, 6 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 x-scale (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 16 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) x-scale)) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 0 points increase in error, 31 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))) x-scale) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 1 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) y-scale)) (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) x-scale) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 30 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (*.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale)) y-scale) (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) x-scale) (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))): 0 points increase in error, 31 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) (/.f64 x-scale (/.f64 (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))) y-scale))))): 31 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (/.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) y-scale)))): 0 points increase in error, 31 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (/.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) (Rewrite<= *-commutative_binary64 (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))))))): 31 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (*.f64 y-scale (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))))))): 0 points increase in error, 17 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (/.f64 x-scale (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64))))) y-scale)))): 17 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale)) y-scale))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale) (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4)) (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale)))): 0 points increase in error, 31 points decrease in error
      (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) 4) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale))) (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (neg.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)) (*.f64 4 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale)))) (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale))): 31 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale) (*.f64 4 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale))))) (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale))): 0 points increase in error, 0 points decrease in error
      (+.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 4 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)))) (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale)) (neg.f64 (*.f64 (*.f64 4 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale))))): 0 points increase in error, 31 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 4 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2)) y-scale) y-scale)))): 31 points increase in error, 0 points decrease in error

   3. Taylor expanded in angle around 0 35.0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   4. Simplified 16.1

      \[\leadsto \color{blue}{\left(-4 \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \]
      Proof
      (*.f64 (*.f64 -4 (*.f64 (/.f64 a x-scale) (/.f64 a x-scale))) (*.f64 (/.f64 b y-scale) (/.f64 b y-scale))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a a) (*.f64 x-scale x-scale)))) (*.f64 (/.f64 b y-scale) (/.f64 b y-scale))): 5 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -4 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 x-scale x-scale))) (*.f64 (/.f64 b y-scale) (/.f64 b y-scale))): 5 points increase in error, 4 points decrease in error
      (*.f64 (*.f64 -4 (/.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2)))) (*.f64 (/.f64 b y-scale) (/.f64 b y-scale))): 0 points increase in error, 5 points decrease in error
      (*.f64 (*.f64 -4 (/.f64 (pow.f64 a 2) (pow.f64 x-scale 2))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 b b) (*.f64 y-scale y-scale)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -4 (/.f64 (pow.f64 a 2) (pow.f64 x-scale 2))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 y-scale y-scale))): 1 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -4 (/.f64 (pow.f64 a 2) (pow.f64 x-scale 2))) (/.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2)))): 4 points increase in error, 1 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -4 (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 x-scale 2)) (/.f64 (pow.f64 b 2) (pow.f64 y-scale 2))))): 0 points increase in error, 4 points decrease in error
      (*.f64 -4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (*.f64 (pow.f64 x-scale 2) (pow.f64 y-scale 2))))): 5 points increase in error, 0 points decrease in error

   5. Taylor expanded in a around 0 35.0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   6. Simplified 4.3

      \[\leadsto \color{blue}{-4 \cdot {\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)}^{2}} \]
      Proof
      (*.f64 -4 (pow.f64 (*.f64 (/.f64 a x-scale) (/.f64 b y-scale)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 (/.f64 a x-scale) (/.f64 b y-scale)) (*.f64 (/.f64 a x-scale) (/.f64 b y-scale))))): 8 points increase in error, 0 points decrease in error
      (*.f64 -4 (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 (/.f64 a x-scale) (/.f64 a x-scale)) (*.f64 (/.f64 b y-scale) (/.f64 b y-scale))))): 5 points increase in error, 6 points decrease in error
      (*.f64 -4 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a a) (*.f64 x-scale x-scale))) (*.f64 (/.f64 b y-scale) (/.f64 b y-scale)))): 0 points increase in error, 5 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 x-scale x-scale)) (*.f64 (/.f64 b y-scale) (/.f64 b y-scale)))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2))) (*.f64 (/.f64 b y-scale) (/.f64 b y-scale)))): 3 points increase in error, 0 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 x-scale 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 b b) (*.f64 y-scale y-scale))))): 7 points increase in error, 1 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 x-scale 2)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 y-scale y-scale)))): 0 points increase in error, 7 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 x-scale 2)) (/.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2))))): 5 points increase in error, 0 points decrease in error
      (*.f64 -4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (*.f64 (pow.f64 x-scale 2) (pow.f64 y-scale 2))))): 3 points increase in error, 0 points decrease in error
      (*.f64 -4 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2))))): 1 points increase in error, 0 points decrease in error

   7. Applied egg-rr 4.4

      \[\leadsto -4 \cdot \color{blue}{\frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{x-scale}{a} \cdot \frac{y-scale}{b}}} \]

   8. Applied egg-rr 4.2

      \[\leadsto -4 \cdot \frac{\color{blue}{\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}}}{\frac{x-scale}{a} \cdot \frac{y-scale}{b}} \]

   if -7.5000000000000002e-60 < x-scale

   1. Initial program 42.5

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

   2. Taylor expanded in angle around 0 41.6

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   3. Simplified 20.7

      \[\leadsto \color{blue}{\left(\frac{a}{y-scale} \cdot \frac{a}{y-scale}\right) \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot -4\right)} \]
      Proof
      (*.f64 (*.f64 (/.f64 a y-scale) (/.f64 a y-scale)) (*.f64 (*.f64 (/.f64 b x-scale) (/.f64 b x-scale)) -4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a a) (*.f64 y-scale y-scale))) (*.f64 (*.f64 (/.f64 b x-scale) (/.f64 b x-scale)) -4)): 6 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 y-scale y-scale)) (*.f64 (*.f64 (/.f64 b x-scale) (/.f64 b x-scale)) -4)): 6 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2))) (*.f64 (*.f64 (/.f64 b x-scale) (/.f64 b x-scale)) -4)): 0 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 y-scale 2)) (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 b b) (*.f64 x-scale x-scale))) -4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 y-scale 2)) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 x-scale x-scale)) -4)): 8 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 y-scale 2)) (*.f64 (/.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2))) -4)): 2 points increase in error, 8 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 y-scale 2)) (/.f64 (pow.f64 b 2) (pow.f64 x-scale 2))) -4)): 0 points increase in error, 10 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2)))) -4): 9 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -4 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2))))): 0 points increase in error, 3 points decrease in error

   4. Taylor expanded in a around 0 41.6

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   5. Simplified 6.4

      \[\leadsto \color{blue}{-4 \cdot {\left(\frac{\frac{a}{y-scale}}{\frac{x-scale}{b}}\right)}^{2}} \]
      Proof
      (*.f64 -4 (pow.f64 (/.f64 (/.f64 a y-scale) (/.f64 x-scale b)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (pow.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 a y-scale) b) x-scale)) 2)): 9 points increase in error, 0 points decrease in error
      (*.f64 -4 (pow.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 a y-scale) (/.f64 b x-scale))) 2)): 5 points increase in error, 4 points decrease in error
      (*.f64 -4 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 (/.f64 a y-scale) (/.f64 b x-scale)) (*.f64 (/.f64 a y-scale) (/.f64 b x-scale))))): 0 points increase in error, 5 points decrease in error
      (*.f64 -4 (Rewrite=> swap-sqr_binary64 (*.f64 (*.f64 (/.f64 a y-scale) (/.f64 a y-scale)) (*.f64 (/.f64 b x-scale) (/.f64 b x-scale))))): 0 points increase in error, 0 points decrease in error
      (*.f64 -4 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a a) (*.f64 y-scale y-scale))) (*.f64 (/.f64 b x-scale) (/.f64 b x-scale)))): 9 points increase in error, 0 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 y-scale y-scale)) (*.f64 (/.f64 b x-scale) (/.f64 b x-scale)))): 2 points increase in error, 9 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 y-scale 2))) (*.f64 (/.f64 b x-scale) (/.f64 b x-scale)))): 0 points increase in error, 9 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 y-scale 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 b b) (*.f64 x-scale x-scale))))): 8 points increase in error, 0 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 y-scale 2)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 x-scale x-scale)))): 0 points increase in error, 3 points decrease in error
      (*.f64 -4 (*.f64 (/.f64 (pow.f64 a 2) (pow.f64 y-scale 2)) (/.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 x-scale 2))))): 6 points increase in error, 0 points decrease in error
      (*.f64 -4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (*.f64 (pow.f64 y-scale 2) (pow.f64 x-scale 2))))): 2 points increase in error, 7 points decrease in error
      (*.f64 -4 (/.f64 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 x-scale 2) (pow.f64 y-scale 2))))): 0 points increase in error, 12 points decrease in error

   6. Applied egg-rr 6.6

      \[\leadsto -4 \cdot \color{blue}{\frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -7.5 \cdot 10^{-60}:\\ \;\;\;\;-4 \cdot \frac{\frac{\frac{a}{x-scale}}{\frac{y-scale}{b}}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.5
Cost	1617

\[\begin{array}{l} t_0 := -4 \cdot \frac{\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot b\right)}{y-scale}\\ t_1 := \frac{a \cdot b}{x-scale \cdot y-scale}\\ \mathbf{if}\;x-scale \leq -1.75 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -4.2 \cdot 10^{-154}:\\ \;\;\;\;-4 \cdot \left(\frac{t_1}{x-scale} \cdot \frac{a}{\frac{y-scale}{b}}\right)\\ \mathbf{elif}\;x-scale \leq 9.8 \cdot 10^{-45} \lor \neg \left(x-scale \leq 3.4 \cdot 10^{+144}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{t_1}{\frac{x-scale \cdot y-scale}{a}}\right)\\ \end{array} \]

Alternative 2
Error	9.6
Cost	1616

\[\begin{array}{l} t_0 := -4 \cdot \frac{\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot b\right)}{y-scale}\\ t_1 := \frac{a \cdot b}{x-scale \cdot y-scale}\\ \mathbf{if}\;x-scale \leq -1.5 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x-scale \leq -3.5 \cdot 10^{-154}:\\ \;\;\;\;-4 \cdot \left(\frac{t_1}{x-scale} \cdot \frac{a}{\frac{y-scale}{b}}\right)\\ \mathbf{elif}\;x-scale \leq 6.2 \cdot 10^{-98}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)}{y-scale \cdot \frac{x-scale}{b}}\\ \mathbf{elif}\;x-scale \leq 7.3 \cdot 10^{+143}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{t_1}{\frac{x-scale \cdot y-scale}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.6
Cost	1485

\[\begin{array}{l} \mathbf{if}\;angle \leq -2.4 \cdot 10^{-274} \lor \neg \left(angle \leq 1.28 \cdot 10^{-21}\right) \land angle \leq 1.55 \cdot 10^{+153}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a \cdot b}{x-scale \cdot y-scale}}{\frac{x-scale \cdot y-scale}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{y-scale \cdot \left(\frac{x-scale}{b} \cdot \frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}\right)}\\ \end{array} \]

Alternative 4
Error	11.1
Cost	1485

\[\begin{array}{l} \mathbf{if}\;angle \leq -6.5 \cdot 10^{-275}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a \cdot b}{x-scale \cdot y-scale}}{\frac{x-scale \cdot y-scale}{a}}\right)\\ \mathbf{elif}\;angle \leq 1.95 \cdot 10^{-10} \lor \neg \left(angle \leq 5 \cdot 10^{+159}\right):\\ \;\;\;\;-4 \cdot \frac{a}{y-scale \cdot \left(\frac{x-scale}{b} \cdot \frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a \cdot b}{\frac{x-scale \cdot y-scale}{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}}\\ \end{array} \]

Alternative 5
Error	11.3
Cost	1484

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-259}:\\ \;\;\;\;-4 \cdot \frac{a}{y-scale \cdot \left(\frac{x-scale}{b} \cdot \frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}\right)}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-178}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a \cdot b}{x-scale \cdot y-scale}}{x-scale} \cdot \frac{a}{\frac{y-scale}{b}}\right)\\ \mathbf{elif}\;b \leq 7.3 \cdot 10^{+146}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \left(b \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(a \cdot b\right)}{x-scale \cdot y-scale}\\ \end{array} \]

Alternative 6
Error	24.8
Cost	1353

\[\begin{array}{l} \mathbf{if}\;y-scale \leq -6.5 \cdot 10^{-139} \lor \neg \left(y-scale \leq 4 \cdot 10^{-127}\right):\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{\frac{b}{x-scale}}{x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	6.7
Cost	1353

\[\begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{-175} \lor \neg \left(b \leq 7.4 \cdot 10^{+140}\right):\\ \;\;\;\;-4 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \left(b \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	12.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{-175}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{a \cdot b}{x-scale \cdot y-scale}}{x-scale} \cdot \frac{a}{\frac{y-scale}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \left(b \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	10.8
Cost	1220

\[\begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-249}:\\ \;\;\;\;-4 \cdot \left(b \cdot \frac{\frac{a \cdot b}{x-scale \cdot y-scale}}{\frac{x-scale \cdot y-scale}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \left(b \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	5.9
Cost	1220

\[\begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-266}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{x-scale} \cdot \frac{b}{y-scale}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a}}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{y-scale} \cdot \frac{b}{x-scale}}{\frac{\frac{x-scale}{b}}{\frac{a}{y-scale}}}\\ \end{array} \]

Alternative 11
Error	12.7
Cost	1088

\[-4 \cdot \left(\frac{a}{x-scale \cdot y-scale} \cdot \left(b \cdot \left(\frac{a}{y-scale} \cdot \frac{b}{x-scale}\right)\right)\right) \]

Alternative 12
Error	30.3
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 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 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (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))))