Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.4% → 93.8%

Time: 2.0min

Alternatives: 3

Speedup: 1693.0×

• 33.9% 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: 24.4% 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: 93.8% accurate, 15.4× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (pow (/ (* a b) (* y-scale x-scale)) 2.0)))

C

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

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 = (-4.0d0) * (((a * b) / (y_45scale * x_45scale)) ** 2.0d0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * Math.pow(((a * b) / (y_45_scale * x_45_scale)), 2.0);
}

Python

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

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(-4.0 * (Float64(Float64(a * b) / Float64(y_45_scale * x_45_scale)) ^ 2.0))
end

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[Power[N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.0%

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

3. Simplified 23.0%

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

5. Taylor expanded in angle around 0 48.0%

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

   1. *-commutative 48.0%

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

7. Simplified 48.0%

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

   1. add-exp-log 47.9%

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

   2. div-inv 47.6%

      \[\leadsto -4 \cdot e^{\log \color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}} \]

   3. *-commutative 47.6%

      \[\leadsto -4 \cdot e^{\log \left(\color{blue}{\left({a}^{2} \cdot {b}^{2}\right)} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

   4. pow-prod-down 58.8%

      \[\leadsto -4 \cdot e^{\log \left(\color{blue}{{\left(a \cdot b\right)}^{2}} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

   5. pow-prod-down 76.6%

      \[\leadsto -4 \cdot e^{\log \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)} \]

   6. *-commutative 76.6%

      \[\leadsto -4 \cdot e^{\log \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right)} \]

   7. pow-flip 77.0%

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

   8. *-commutative 77.0%

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

   9. metadata-eval 77.0%

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

9. Applied egg-rr 77.0%

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

    1. log-prod 76.6%

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

    2. log-pow 46.3%

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

11. Applied egg-rr 46.3%

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

13. Applied egg-rr 68.0%

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

    1. expm1-def 74.6%

       \[\leadsto -4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{a \cdot b}{x-scale}\right)}^{2} \cdot {y-scale}^{-2}\right)\right)} \]

    2. expm1-log1p 75.1%

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

    3. unpow2 75.1%

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

    4. sqr-pow 75.1%

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

    5. unswap-sqr 93.4%

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

    6. metadata-eval 93.4%

       \[\leadsto -4 \cdot \left(\left(\frac{a \cdot b}{x-scale} \cdot {y-scale}^{\color{blue}{-1}}\right) \cdot \left(\frac{a \cdot b}{x-scale} \cdot {y-scale}^{\left(\frac{-2}{2}\right)}\right)\right) \]

    7. unpow-1 93.4%

       \[\leadsto -4 \cdot \left(\left(\frac{a \cdot b}{x-scale} \cdot \color{blue}{\frac{1}{y-scale}}\right) \cdot \left(\frac{a \cdot b}{x-scale} \cdot {y-scale}^{\left(\frac{-2}{2}\right)}\right)\right) \]

    8. times-frac 92.1%

       \[\leadsto -4 \cdot \left(\color{blue}{\frac{\left(a \cdot b\right) \cdot 1}{x-scale \cdot y-scale}} \cdot \left(\frac{a \cdot b}{x-scale} \cdot {y-scale}^{\left(\frac{-2}{2}\right)}\right)\right) \]

    9. associate-*r/ 92.1%

       \[\leadsto -4 \cdot \left(\color{blue}{\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right)} \cdot \left(\frac{a \cdot b}{x-scale} \cdot {y-scale}^{\left(\frac{-2}{2}\right)}\right)\right) \]

    10. metadata-eval 92.1%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \left(\frac{a \cdot b}{x-scale} \cdot {y-scale}^{\color{blue}{-1}}\right)\right) \]

    11. unpow-1 92.1%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \left(\frac{a \cdot b}{x-scale} \cdot \color{blue}{\frac{1}{y-scale}}\right)\right) \]

    12. times-frac 93.8%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \color{blue}{\frac{\left(a \cdot b\right) \cdot 1}{x-scale \cdot y-scale}}\right) \]

    13. associate-*r/ 93.7%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \color{blue}{\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right)}\right) \]

    14. unpow2 93.7%

        \[\leadsto -4 \cdot \color{blue}{{\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right)}^{2}} \]

    15. associate-*r/ 93.8%

        \[\leadsto -4 \cdot {\color{blue}{\left(\frac{\left(a \cdot b\right) \cdot 1}{x-scale \cdot y-scale}\right)}}^{2} \]

    16. *-rgt-identity 93.8%

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

    17. *-commutative 93.8%

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

15. Simplified 93.8%

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

16. Final simplification 93.8%

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

Alternative 2: 93.7% accurate, 80.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot b\right) \cdot \frac{1}{y-scale \cdot x-scale}\\ -4 \cdot \left(t_0 \cdot t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* a b) (/ 1.0 (* y-scale x-scale))))) (* -4.0 (* t_0 t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a * b) * (1.0 / (y_45_scale * x_45_scale));
	return -4.0 * (t_0 * t_0);
}

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
    real(8) :: t_0
    t_0 = (a * b) * (1.0d0 / (y_45scale * x_45scale))
    code = (-4.0d0) * (t_0 * t_0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a * b) * (1.0 / (y_45_scale * x_45_scale));
	return -4.0 * (t_0 * t_0);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a * b) * (1.0 / (y_45_scale * x_45_scale))
	return -4.0 * (t_0 * t_0)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a * b) * Float64(1.0 / Float64(y_45_scale * x_45_scale)))
	return Float64(-4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a * b) * (1.0 / (y_45_scale * x_45_scale));
	tmp = -4.0 * (t_0 * t_0);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] * N[(1.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 25.0%

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

3. Simplified 23.0%

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

5. Taylor expanded in angle around 0 48.0%

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

   1. *-commutative 48.0%

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

7. Simplified 48.0%

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

   1. add-exp-log 47.9%

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

   2. div-inv 47.6%

      \[\leadsto -4 \cdot e^{\log \color{blue}{\left(\left({b}^{2} \cdot {a}^{2}\right) \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)}} \]

   3. *-commutative 47.6%

      \[\leadsto -4 \cdot e^{\log \left(\color{blue}{\left({a}^{2} \cdot {b}^{2}\right)} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

   4. pow-prod-down 58.8%

      \[\leadsto -4 \cdot e^{\log \left(\color{blue}{{\left(a \cdot b\right)}^{2}} \cdot \frac{1}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]

   5. pow-prod-down 76.6%

      \[\leadsto -4 \cdot e^{\log \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{\color{blue}{{\left(x-scale \cdot y-scale\right)}^{2}}}\right)} \]

   6. *-commutative 76.6%

      \[\leadsto -4 \cdot e^{\log \left({\left(a \cdot b\right)}^{2} \cdot \frac{1}{{\color{blue}{\left(y-scale \cdot x-scale\right)}}^{2}}\right)} \]

   7. pow-flip 77.0%

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

   8. *-commutative 77.0%

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

   9. metadata-eval 77.0%

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

9. Applied egg-rr 77.0%

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

    1. log-prod 76.6%

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

    2. log-pow 46.3%

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

11. Applied egg-rr 46.3%

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

    1. exp-sum 44.5%

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

    2. *-commutative 44.5%

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

    3. pow-to-exp 77.0%

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

    4. add-exp-log 78.1%

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

    5. add-sqr-sqrt 78.1%

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

    6. sqrt-prod 78.1%

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

    7. pow2 78.1%

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

    8. sqrt-prod 45.2%

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

    9. add-sqr-sqrt 56.3%

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

    10. sqrt-pow1 55.2%

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

    11. metadata-eval 55.2%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-1}}\right) \cdot \sqrt{{\left(a \cdot b\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}}\right) \]

    12. unpow-1 55.2%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \color{blue}{\frac{1}{x-scale \cdot y-scale}}\right) \cdot \sqrt{{\left(a \cdot b\right)}^{2} \cdot {\left(x-scale \cdot y-scale\right)}^{-2}}\right) \]

    13. sqrt-prod 55.2%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \color{blue}{\left(\sqrt{{\left(a \cdot b\right)}^{2}} \cdot \sqrt{{\left(x-scale \cdot y-scale\right)}^{-2}}\right)}\right) \]

    14. pow2 55.2%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \left(\sqrt{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}} \cdot \sqrt{{\left(x-scale \cdot y-scale\right)}^{-2}}\right)\right) \]

    15. sqrt-prod 33.6%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \left(\color{blue}{\left(\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}\right)} \cdot \sqrt{{\left(x-scale \cdot y-scale\right)}^{-2}}\right)\right) \]

    16. add-sqr-sqrt 58.6%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \left(\color{blue}{\left(a \cdot b\right)} \cdot \sqrt{{\left(x-scale \cdot y-scale\right)}^{-2}}\right)\right) \]

    17. sqrt-pow1 93.7%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \left(\left(a \cdot b\right) \cdot \color{blue}{{\left(x-scale \cdot y-scale\right)}^{\left(\frac{-2}{2}\right)}}\right)\right) \]

    18. metadata-eval 93.7%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \left(\left(a \cdot b\right) \cdot {\left(x-scale \cdot y-scale\right)}^{\color{blue}{-1}}\right)\right) \]

    19. unpow-1 93.7%

        \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \left(\left(a \cdot b\right) \cdot \color{blue}{\frac{1}{x-scale \cdot y-scale}}\right)\right) \]

13. Applied egg-rr 93.7%

    \[\leadsto -4 \cdot \color{blue}{\left(\left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right) \cdot \left(\left(a \cdot b\right) \cdot \frac{1}{x-scale \cdot y-scale}\right)\right)} \]

14. Final simplification 93.7%

    \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \frac{1}{y-scale \cdot x-scale}\right) \cdot \left(\left(a \cdot b\right) \cdot \frac{1}{y-scale \cdot x-scale}\right)\right) \]
15. Add Preprocessing

Alternative 3: 35.3% accurate, 1693.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale) :precision binary64 0.0)

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return 0.0
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 25.0%

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

3. Simplified 24.7%

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

5. Taylor expanded in b around 0 24.4%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + 4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
6. Step-by-step derivation

   1. distribute-rgt-out 24.4%

      \[\leadsto \color{blue}{\frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \left(-4 + 4\right)} \]

   2. metadata-eval 24.4%

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{0} \]

   3. mul0-rgt 34.8%

      \[\leadsto \color{blue}{0} \]

7. Simplified 34.8%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 34.8%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))))