Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.7% → 92.6%

Time: 27.9s

Alternatives: 1

Speedup: 99.6×

• 34.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 24.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 92.6% accurate, 99.6× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* b (/ a (/ (* x-scale y-scale) -4.0))) (* a (/ b (* x-scale y-scale)))))

C

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

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (b * (a / ((x_45scale * y_45scale) / (-4.0d0)))) * (a * (b / (x_45scale * y_45scale)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (b * (a / ((x_45_scale * y_45_scale) / -4.0))) * (a * (b / (x_45_scale * y_45_scale)));
end

Wolfram

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

Derivation

1. Initial program 19.2%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{{\color{blue}{x-scale}}^{2}}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2} \cdot {a}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{b}^{2}}{x-scale} \cdot \color{blue}{\frac{{a}^{2}}{x-scale}}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{b}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{a}^{2}}{x-scale}\right)}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({b}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{a}^{2}}}{x-scale}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \left(\frac{{\color{blue}{a}}^{2}}{x-scale}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left({a}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right)\right)\right) \]

   17. *-lowering-*.f64 54.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right)\right)\right) \]

5. Simplified 54.5%

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

   1. *-commutative N/A

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

   2. frac-times N/A

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

   3. frac-times N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right) \cdot -4\right), \color{blue}{\left(\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right), -4\right), \left(\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot y-scale\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right), -4\right), \left(\left(\color{blue}{x-scale} \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot \left(a \cdot a\right)\right)\right), -4\right), \left(\left(\color{blue}{x-scale} \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(a \cdot a\right)\right)\right), -4\right), \left(\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right)\right), -4\right), \left(\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)\right)\right) \]

   10. unswap-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right)\right), -4\right), \left(\left(x-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot y-scale\right)}\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right)\right), -4\right), \left(x-scale \cdot \color{blue}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right)\right), -4\right), \mathsf{*.f64}\left(x-scale, \color{blue}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right)\right), -4\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, \color{blue}{\left(x-scale \cdot y-scale\right)}\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right)\right), -4\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, \left(y-scale \cdot \color{blue}{x-scale}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 68.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right)\right), -4\right), \mathsf{*.f64}\left(x-scale, \mathsf{*.f64}\left(y-scale, \mathsf{*.f64}\left(y-scale, \color{blue}{x-scale}\right)\right)\right)\right) \]

7. Applied egg-rr 68.2%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{b}{y-scale \cdot x-scale}\right), \color{blue}{\left(\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot -4}{y-scale \cdot x-scale}\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \left(y-scale \cdot x-scale\right)\right), \left(\frac{\color{blue}{\left(b \cdot \left(a \cdot a\right)\right) \cdot -4}}{y-scale \cdot x-scale}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \left(x-scale \cdot y-scale\right)\right), \left(\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{-4}}{y-scale \cdot x-scale}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(x-scale, y-scale\right)\right), \left(\frac{\left(b \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{-4}}{y-scale \cdot x-scale}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(x-scale, y-scale\right)\right), \mathsf{/.f64}\left(\left(\left(b \cdot \left(a \cdot a\right)\right) \cdot -4\right), \color{blue}{\left(y-scale \cdot x-scale\right)}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(x-scale, y-scale\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(b \cdot \left(a \cdot a\right)\right), -4\right), \left(\color{blue}{y-scale} \cdot x-scale\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(x-scale, y-scale\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(a \cdot a\right)\right), -4\right), \left(y-scale \cdot x-scale\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(x-scale, y-scale\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right), -4\right), \left(y-scale \cdot x-scale\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(x-scale, y-scale\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right), -4\right), \left(x-scale \cdot \color{blue}{y-scale}\right)\right)\right) \]

   14. *-lowering-*.f64 77.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(b, \mathsf{*.f64}\left(x-scale, y-scale\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, a\right)\right), -4\right), \mathsf{*.f64}\left(x-scale, \color{blue}{y-scale}\right)\right)\right) \]

9. Applied egg-rr 77.3%

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

    1. frac-times N/A

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

    2. associate-*r* N/A

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

    3. associate-*l* N/A

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

    4. swap-sqr N/A

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

    5. associate-*l* N/A

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

    6. times-frac N/A

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

    7. *-commutative N/A

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

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(b \cdot a\right) \cdot -4}{x-scale \cdot y-scale}\right), \color{blue}{\left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)}\right) \]

    9. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{b \cdot \left(a \cdot -4\right)}{x-scale \cdot y-scale}\right), \left(\frac{\color{blue}{b} \cdot a}{x-scale \cdot y-scale}\right)\right) \]

    10. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(\left(b \cdot \frac{a \cdot -4}{x-scale \cdot y-scale}\right), \left(\frac{\color{blue}{b \cdot a}}{x-scale \cdot y-scale}\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{a \cdot -4}{x-scale \cdot y-scale}\right)\right), \left(\frac{\color{blue}{b \cdot a}}{x-scale \cdot y-scale}\right)\right) \]

    12. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right)\right), \left(\frac{b \cdot \color{blue}{a}}{x-scale \cdot y-scale}\right)\right) \]

    13. clear-num N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(a \cdot \frac{1}{\frac{x-scale \cdot y-scale}{-4}}\right)\right), \left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)\right) \]

    14. un-div-inv N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{a}{\frac{x-scale \cdot y-scale}{-4}}\right)\right), \left(\frac{b \cdot \color{blue}{a}}{x-scale \cdot y-scale}\right)\right) \]

    15. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(a, \left(\frac{x-scale \cdot y-scale}{-4}\right)\right)\right), \left(\frac{b \cdot \color{blue}{a}}{x-scale \cdot y-scale}\right)\right) \]

    16. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), -4\right)\right)\right), \left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)\right) \]

    17. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), -4\right)\right)\right), \left(\frac{b \cdot a}{x-scale \cdot y-scale}\right)\right) \]

    18. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), -4\right)\right)\right), \left(\frac{a \cdot b}{\color{blue}{x-scale} \cdot y-scale}\right)\right) \]

    19. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), -4\right)\right)\right), \left(a \cdot \color{blue}{\frac{b}{x-scale \cdot y-scale}}\right)\right) \]

    20. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), -4\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{b}{x-scale \cdot y-scale}\right)}\right)\right) \]

    21. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), -4\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(b, \color{blue}{\left(x-scale \cdot y-scale\right)}\right)\right)\right) \]

11. Applied egg-rr 91.1%

    \[\leadsto \color{blue}{\left(b \cdot \frac{a}{\frac{x-scale \cdot y-scale}{-4}}\right) \cdot \left(a \cdot \frac{b}{x-scale \cdot y-scale}\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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))))