Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.4% → 93.8%

Time: 2.3min

Precision: binary64

Cost: 7040

• 32.7% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \\ -4 \cdot {\left(\frac{a}{\frac{y-scale}{b} \cdot x-scale}\right)}^{2} \end{array} \]

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
 (* -4.0 (pow (/ a (* (/ y-scale b) x-scale)) 2.0)))

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) {
	return -4.0 * pow((a / ((y_45_scale / b) * x_45_scale)), 2.0);
}

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) {
	return -4.0 * Math.pow((a / ((y_45_scale / b) * x_45_scale)), 2.0);
}

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):
	return -4.0 * math.pow((a / ((y_45_scale / b) * x_45_scale)), 2.0)

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)
	return Float64(-4.0 * (Float64(a / Float64(Float64(y_45_scale / b) * x_45_scale)) ^ 2.0))
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 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = -4.0 * ((a / ((y_45_scale / b) * x_45_scale)) ^ 2.0);
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_] := N[(-4.0 * N[Power[N[(a / N[(N[(y$45$scale / b), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.4%

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

2. Simplified 22.1%

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

   [Start] 21.4%	\[ \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   cancel-sign-sub-inv [=>] 21.4%	\[ \color{blue}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} + \left(-4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}} \]
   fma-def [=>] 25.5%	\[ \color{blue}{\mathsf{fma}\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}, \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}, \left(-4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)} \]

3. Taylor expanded in angle around 0 53.0%

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

4. Simplified 50.9%

   \[\leadsto \color{blue}{-4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right)} \]
   Step-by-step derivation

   [Start] 53.0%	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
   times-frac [=>] 50.9%	\[ -4 \cdot \color{blue}{\left(\frac{{a}^{2}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)} \]
   unpow2 [=>] 50.9%	\[ -4 \cdot \left(\frac{\color{blue}{a \cdot a}}{{x-scale}^{2}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
   unpow2 [=>] 50.9%	\[ -4 \cdot \left(\frac{a \cdot a}{\color{blue}{x-scale \cdot x-scale}} \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right) \]
   unpow2 [=>] 50.9%	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{\color{blue}{b \cdot b}}{{y-scale}^{2}}\right) \]
   unpow2 [=>] 50.9%	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{\color{blue}{y-scale \cdot y-scale}}\right) \]

5. Applied egg-rr 71.1%

   \[\leadsto -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{b}{y-scale}\right)}^{2} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right)} - 1\right)} \]
   Step-by-step derivation

   [Start] 50.9%	\[ -4 \cdot \left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
   expm1-log1p-u [=>] 50.7%	\[ -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right)\right)} \]
   expm1-udef [=>] 50.1%	\[ -4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right)} - 1\right)} \]
   *-commutative [=>] 50.1%	\[ -4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{b \cdot b}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale \cdot x-scale}}\right)} - 1\right) \]
   times-frac [=>] 59.6%	\[ -4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)} \cdot \frac{a \cdot a}{x-scale \cdot x-scale}\right)} - 1\right) \]
   pow2 [=>] 59.6%	\[ -4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{b}{y-scale}\right)}^{2}} \cdot \frac{a \cdot a}{x-scale \cdot x-scale}\right)} - 1\right) \]
   times-frac [=>] 71.1%	\[ -4 \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{b}{y-scale}\right)}^{2} \cdot \color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)}\right)} - 1\right) \]
   pow2 [=>] 71.1%	\[ -4 \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{b}{y-scale}\right)}^{2} \cdot \color{blue}{{\left(\frac{a}{x-scale}\right)}^{2}}\right)} - 1\right) \]

6. Simplified 78.4%

   \[\leadsto -4 \cdot \color{blue}{\left({\left(\frac{b}{y-scale}\right)}^{2} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right)} \]
   Step-by-step derivation

   [Start] 71.1%	\[ -4 \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{b}{y-scale}\right)}^{2} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right)} - 1\right) \]
   expm1-def [=>] 77.7%	\[ -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{b}{y-scale}\right)}^{2} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right)\right)} \]
   expm1-log1p [=>] 78.4%	\[ -4 \cdot \color{blue}{\left({\left(\frac{b}{y-scale}\right)}^{2} \cdot {\left(\frac{a}{x-scale}\right)}^{2}\right)} \]

7. Taylor expanded in b around 0 53.0%

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

8. Simplified 95.3%

   \[\leadsto -4 \cdot \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \]
   Step-by-step derivation

   [Start] 53.0%	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]
   unpow2 [=>] 53.0%	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]
   unpow2 [=>] 53.0%	\[ -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\left(x-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}} \]
   unpow2 [=>] 53.0%	\[ -4 \cdot \frac{\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} \]
   unpow2 [=>] 53.0%	\[ -4 \cdot \frac{\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot b\right)}}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} \]
   times-frac [=>] 50.9%	\[ -4 \cdot \color{blue}{\left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right)} \]
   times-frac [=>] 61.4%	\[ -4 \cdot \left(\color{blue}{\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)} \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right) \]
   times-frac [=>] 78.4%	\[ -4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)}\right) \]
   swap-sqr [<=] 95.3%	\[ -4 \cdot \color{blue}{\left(\left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right)} \]
   *-commutative [<=] 95.3%	\[ -4 \cdot \left(\color{blue}{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)} \cdot \left(\frac{a}{x-scale} \cdot \frac{b}{y-scale}\right)\right) \]
   *-commutative [<=] 95.3%	\[ -4 \cdot \left(\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right) \cdot \color{blue}{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}\right) \]
   unpow2 [<=] 95.3%	\[ -4 \cdot \color{blue}{{\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2}} \]

9. Applied egg-rr 96.1%

   \[\leadsto -4 \cdot {\color{blue}{\left(\frac{a}{\frac{y-scale}{b} \cdot x-scale}\right)}}^{2} \]
   Step-by-step derivation

   [Start] 95.3%	\[ -4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2} \]
   clear-num [=>] 94.9%	\[ -4 \cdot {\left(\color{blue}{\frac{1}{\frac{y-scale}{b}}} \cdot \frac{a}{x-scale}\right)}^{2} \]
   frac-times [=>] 96.1%	\[ -4 \cdot {\color{blue}{\left(\frac{1 \cdot a}{\frac{y-scale}{b} \cdot x-scale}\right)}}^{2} \]
   *-un-lft-identity [<=] 96.1%	\[ -4 \cdot {\left(\frac{\color{blue}{a}}{\frac{y-scale}{b} \cdot x-scale}\right)}^{2} \]

10. Final simplification 96.1%

    \[\leadsto -4 \cdot {\left(\frac{a}{\frac{y-scale}{b} \cdot x-scale}\right)}^{2} \]

Alternatives

Alternative 1
Accuracy	93.8%
Cost	7040

\[-4 \cdot {\left(\frac{a}{\frac{y-scale}{b} \cdot x-scale}\right)}^{2} \]

Alternative 2
Accuracy	93.9%
Cost	7040

\[-4 \cdot {\left(\frac{b}{y-scale} \cdot \frac{a}{x-scale}\right)}^{2} \]

Alternative 3
Accuracy	93.3%
Cost	1220

\[\begin{array}{l} t_0 := \frac{a \cdot b}{y-scale \cdot x-scale}\\ \mathbf{if}\;b \leq -2.85 \cdot 10^{+135}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{b}{y-scale}\right) \cdot \frac{\frac{a}{x-scale}}{\frac{y-scale}{b} \cdot x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot t_0\right)\\ \end{array} \]

Alternative 4
Accuracy	77.4%
Cost	1088

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

Alternative 5
Accuracy	88.4%
Cost	1088

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

Alternative 6
Accuracy	93.7%
Cost	1088

\[\begin{array}{l} t_0 := \frac{a \cdot \frac{b}{y-scale}}{x-scale}\\ -4 \cdot \left(t_0 \cdot t_0\right) \end{array} \]

Alternative 7
Accuracy	36.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023167 
(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))))