Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.1% → 93.7%

Time: 1.8min

Alternatives: 4

Speedup: 146.2×

• 33.6% 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: 24.1% 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: 93.7% accurate, 22.6× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* -4.0 (pow (/ (* b (/ a 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(((b * (a / 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) * (((b * (a / 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(((b * (a / 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(((b * (a / 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(b * Float64(a / 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 * (((b * (a / 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[(b * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.6%

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

2. Taylor expanded in angle around 0 48.6%

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

   1. times-frac 49.0%

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

   2. associate-*r* 49.0%

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

   3. unpow2 49.0%

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

   4. unpow2 49.0%

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

   5. times-frac 61.5%

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

   6. unpow2 61.5%

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

   7. unpow2 61.5%

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

   8. times-frac 80.1%

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

4. Simplified 80.1%

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

5. Taylor expanded in a around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. times-frac 49.0%

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

   3. unpow2 49.0%

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

   4. unpow2 49.0%

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

   5. times-frac 61.5%

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

   6. unpow2 61.5%

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

   7. unpow2 61.5%

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

   8. unpow2 61.5%

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

   9. times-frac 80.4%

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

   10. unpow2 80.4%

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

7. Simplified 80.4%

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

8. Taylor expanded in a around 0 48.6%

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

   1. unpow2 48.6%

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

   2. unpow2 48.6%

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

   3. *-commutative 48.6%

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

   4. times-frac 49.0%

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

   5. times-frac 61.5%

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

   6. unpow2 61.5%

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

   7. unpow2 61.5%

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

   8. times-frac 80.4%

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

   9. swap-sqr 96.3%

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

   10. unpow2 96.3%

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

   11. *-commutative 96.3%

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

   12. associate-*l/ 96.7%

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

10. Simplified 96.7%

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

11. Final simplification 96.7%

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

Alternative 2: 93.6% accurate, 22.6× speedup

Math

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

FPCore

(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 -4.0 * pow(((a / y_45_scale) * (b / 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 / y_45scale) * (b / 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 / y_45_scale) * (b / x_45_scale)), 2.0);
}

Python

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(-4.0 * (Float64(Float64(a / y_45_scale) * Float64(b / x_45_scale)) ^ 2.0))
end

MATLAB

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[(-4.0 * N[Power[N[(N[(a / y$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.6%

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

2. Taylor expanded in angle around 0 48.6%

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

   1. times-frac 49.0%

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

   2. associate-*r* 49.0%

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

   3. unpow2 49.0%

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

   4. unpow2 49.0%

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

   5. times-frac 61.5%

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

   6. unpow2 61.5%

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

   7. unpow2 61.5%

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

   8. times-frac 80.1%

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

4. Simplified 80.1%

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

5. Taylor expanded in a around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. times-frac 49.0%

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

   3. unpow2 49.0%

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

   4. unpow2 49.0%

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

   5. times-frac 61.5%

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

   6. unpow2 61.5%

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

   7. unpow2 61.5%

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

   8. unpow2 61.5%

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

   9. times-frac 80.4%

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

   10. unpow2 80.4%

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

7. Simplified 80.4%

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

   1. pow-prod-down 96.3%

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

9. Applied egg-rr 96.3%

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

10. Final simplification 96.3%

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

Alternative 3: 46.6% accurate, 146.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * (((a * a) / (y_45_scale * y_45_scale)) * ((b * b) / (x_45_scale * x_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 = (-4.0d0) * (((a * a) / (y_45scale * y_45scale)) * ((b * b) / (x_45scale * x_45scale)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.6%

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

2. Taylor expanded in angle around 0 48.6%

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

   1. times-frac 49.0%

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

   2. unpow2 49.0%

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

   3. unpow2 49.0%

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

   4. unpow2 49.0%

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

   5. unpow2 49.0%

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

4. Simplified 49.0%

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

5. Final simplification 49.0%

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

Alternative 4: 94.0% accurate, 146.2× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b a) (* 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 = (b * a) / (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 = (b * a) / (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 = (b * a) / (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 = (b * a) / (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(b * a) / 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 = (b * a) / (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[(b * a), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.6%

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

2. Taylor expanded in angle around 0 48.6%

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

   1. times-frac 49.0%

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

   2. associate-*r* 49.0%

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

   3. unpow2 49.0%

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

   4. unpow2 49.0%

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

   5. times-frac 61.5%

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

   6. unpow2 61.5%

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

   7. unpow2 61.5%

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

   8. times-frac 80.1%

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

4. Simplified 80.1%

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

5. Taylor expanded in a around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. times-frac 49.0%

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

   3. unpow2 49.0%

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

   4. unpow2 49.0%

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

   5. times-frac 61.5%

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

   6. unpow2 61.5%

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

   7. unpow2 61.5%

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

   8. unpow2 61.5%

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

   9. times-frac 80.4%

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

   10. unpow2 80.4%

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

7. Simplified 80.4%

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

8. Taylor expanded in a around 0 48.6%

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

   1. unpow2 48.6%

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

   2. unpow2 48.6%

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

   3. *-commutative 48.6%

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

   4. times-frac 49.0%

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

   5. times-frac 61.5%

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

   6. unpow2 61.5%

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

   7. unpow2 61.5%

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

   8. times-frac 80.4%

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

   9. swap-sqr 96.3%

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

   10. unpow2 96.3%

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

   11. *-commutative 96.3%

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

   12. associate-*l/ 96.7%

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

10. Simplified 96.7%

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

    1. *-commutative 96.7%

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

    2. associate-*r/ 96.3%

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

    3. unpow2 96.3%

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

    4. frac-times 92.7%

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

    5. frac-times 96.0%

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

12. Applied egg-rr 96.0%

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

13. Final simplification 96.0%

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

Reproduce

herbie shell --seed 2023187 
(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))))