Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.8% → 84.9%

Time: 2.1min

Alternatives: 5

Speedup: 2485.0×

• 33.7% 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: 24.8% 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: 84.9% accurate, 130.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 6 \cdot 10^{+199}:\\ \;\;\;\;\left(b \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{-4}{x-scale \cdot y-scale} \cdot \left(a \cdot \frac{b}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 6e+199)
   (* (* b (/ a x-scale)) (* (/ -4.0 (* x-scale y-scale)) (* a (/ b y-scale))))
   (*
    -4.0
    (* (* (/ a x-scale) (/ a x-scale)) (* (/ b y-scale) (/ b y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 6e+199) {
		tmp = (b * (a / x_45_scale)) * ((-4.0 / (x_45_scale * y_45_scale)) * (a * (b / y_45_scale)));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	}
	return tmp;
}

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) :: tmp
    if (x_45scale <= 6d+199) then
        tmp = (b * (a / x_45scale)) * (((-4.0d0) / (x_45scale * y_45scale)) * (a * (b / y_45scale)))
    else
        tmp = (-4.0d0) * (((a / x_45scale) * (a / x_45scale)) * ((b / y_45scale) * (b / y_45scale)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 6e+199) {
		tmp = (b * (a / x_45_scale)) * ((-4.0 / (x_45_scale * y_45_scale)) * (a * (b / y_45_scale)));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 6e+199:
		tmp = (b * (a / x_45_scale)) * ((-4.0 / (x_45_scale * y_45_scale)) * (a * (b / y_45_scale)))
	else:
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 6e+199)
		tmp = Float64(Float64(b * Float64(a / x_45_scale)) * Float64(Float64(-4.0 / Float64(x_45_scale * y_45_scale)) * Float64(a * Float64(b / y_45_scale))));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / x_45_scale) * Float64(a / x_45_scale)) * Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= 6e+199)
		tmp = (b * (a / x_45_scale)) * ((-4.0 / (x_45_scale * y_45_scale)) * (a * (b / y_45_scale)));
	else
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 6e+199], N[(N[(b * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(a / x$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 6.0000000000000002e199

   1. Initial program 27.2%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in angle around 0 54.5%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. unpow2 54.5%

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

      4. unpow2 54.5%

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

      5. unpow2 54.5%

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

      6. unpow2 54.5%

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

      7. unswap-sqr 64.1%

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

   6. Simplified 64.1%

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

      1. times-frac 67.4%

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

      2. unswap-sqr 82.4%

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

   8. Applied egg-rr 82.4%

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

      1. times-frac 89.6%

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

   10. Applied egg-rr 89.6%

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

       1. associate-*l/ 90.3%

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

       2. associate-/l* 87.3%

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

   12. Applied egg-rr 87.3%

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

       1. associate-*l/ 86.6%

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

       2. *-commutative 86.6%

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

       3. associate-*l* 87.5%

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

       4. *-commutative 87.5%

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

       5. associate-/l* 88.3%

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

       6. associate-/r/ 86.8%

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

       7. associate-/r/ 88.8%

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

       8. *-commutative 88.8%

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

   14. Simplified 88.8%

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

   if 6.0000000000000002e199 < x-scale

   1. Initial program 37.3%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in angle around 0 41.1%

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

      1. times-frac 37.4%

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

      2. unpow2 37.4%

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

      3. unpow2 37.4%

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

      4. unpow2 37.4%

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

      5. unpow2 37.4%

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

   6. Simplified 37.4%

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

      1. frac-times 60.4%

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

   8. Applied egg-rr 60.4%

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

   9. Taylor expanded in b around 0 60.4%

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

       1. unpow2 60.4%

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

       2. unpow2 60.4%

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

       3. times-frac 88.8%

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

   11. Simplified 88.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 6 \cdot 10^{+199}:\\ \;\;\;\;\left(b \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{-4}{x-scale \cdot y-scale} \cdot \left(a \cdot \frac{b}{y-scale}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 2: 85.0% accurate, 130.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 2.4 \cdot 10^{+200}:\\ \;\;\;\;\left(a \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{-4}{x-scale \cdot y-scale} \cdot \frac{b}{\frac{x-scale}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 2.4e+200)
   (* (* a (/ b y-scale)) (* (/ -4.0 (* x-scale y-scale)) (/ b (/ x-scale a))))
   (*
    -4.0
    (* (* (/ a x-scale) (/ a x-scale)) (* (/ b y-scale) (/ b y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 2.4e+200) {
		tmp = (a * (b / y_45_scale)) * ((-4.0 / (x_45_scale * y_45_scale)) * (b / (x_45_scale / a)));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	}
	return tmp;
}

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) :: tmp
    if (x_45scale <= 2.4d+200) then
        tmp = (a * (b / y_45scale)) * (((-4.0d0) / (x_45scale * y_45scale)) * (b / (x_45scale / a)))
    else
        tmp = (-4.0d0) * (((a / x_45scale) * (a / x_45scale)) * ((b / y_45scale) * (b / y_45scale)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 2.4e+200) {
		tmp = (a * (b / y_45_scale)) * ((-4.0 / (x_45_scale * y_45_scale)) * (b / (x_45_scale / a)));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 2.4e+200:
		tmp = (a * (b / y_45_scale)) * ((-4.0 / (x_45_scale * y_45_scale)) * (b / (x_45_scale / a)))
	else:
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 2.4e+200)
		tmp = Float64(Float64(a * Float64(b / y_45_scale)) * Float64(Float64(-4.0 / Float64(x_45_scale * y_45_scale)) * Float64(b / Float64(x_45_scale / a))));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / x_45_scale) * Float64(a / x_45_scale)) * Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (x_45_scale <= 2.4e+200)
		tmp = (a * (b / y_45_scale)) * ((-4.0 / (x_45_scale * y_45_scale)) * (b / (x_45_scale / a)));
	else
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 2.4e+200], N[(N[(a * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b / N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(a / x$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.4000000000000001e200

   1. Initial program 27.2%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in angle around 0 54.5%

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

      1. associate-*r/ 54.5%

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

      2. *-commutative 54.5%

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

      3. unpow2 54.5%

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

      4. unpow2 54.5%

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

      5. unpow2 54.5%

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

      6. unpow2 54.5%

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

      7. unswap-sqr 64.1%

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

   6. Simplified 64.1%

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

      1. times-frac 67.4%

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

      2. unswap-sqr 82.4%

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

   8. Applied egg-rr 82.4%

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

      1. times-frac 89.6%

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

   10. Applied egg-rr 89.6%

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

       1. associate-*l/ 90.3%

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

       2. associate-/l* 87.3%

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

   12. Applied egg-rr 87.3%

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

       1. associate-*l/ 86.6%

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

       2. associate-*r* 86.3%

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

       3. associate-/l* 86.3%

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

       4. associate-/r/ 88.8%

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

   14. Simplified 88.8%

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

   if 2.4000000000000001e200 < x-scale

   1. Initial program 37.3%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in angle around 0 41.1%

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

      1. times-frac 37.4%

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

      2. unpow2 37.4%

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

      3. unpow2 37.4%

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

      4. unpow2 37.4%

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

      5. unpow2 37.4%

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

   6. Simplified 37.4%

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

      1. frac-times 60.4%

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

   8. Applied egg-rr 60.4%

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

   9. Taylor expanded in b around 0 60.4%

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

       1. unpow2 60.4%

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

       2. unpow2 60.4%

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

       3. times-frac 88.8%

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

   11. Simplified 88.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.4 \cdot 10^{+200}:\\ \;\;\;\;\left(a \cdot \frac{b}{y-scale}\right) \cdot \left(\frac{-4}{x-scale \cdot y-scale} \cdot \frac{b}{\frac{x-scale}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 3: 77.6% accurate, 146.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / 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 = (-4.0d0) * (((a / x_45scale) * (a / x_45scale)) * ((b / y_45scale) * (b / y_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 / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.3%

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

3. Simplified 24.6%

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

4. Taylor expanded in angle around 0 53.1%

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

   1. times-frac 54.2%

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

   2. unpow2 54.2%

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

   3. unpow2 54.2%

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

   4. unpow2 54.2%

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

   5. unpow2 54.2%

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

6. Simplified 54.2%

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

   1. frac-times 64.2%

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

8. Applied egg-rr 64.2%

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

9. Taylor expanded in b around 0 64.2%

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

    1. unpow2 64.2%

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

    2. unpow2 64.2%

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

    3. times-frac 81.6%

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

11. Simplified 81.6%

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

12. Final simplification 81.6%

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

Alternative 4: 88.5% accurate, 146.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 / (x_45_scale * y_45_scale)) * (((b * a) / x_45_scale) * ((b * a) / 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 = ((-4.0d0) / (x_45scale * y_45scale)) * (((b * a) / x_45scale) * ((b * a) / y_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 / (x_45_scale * y_45_scale)) * (((b * a) / x_45_scale) * ((b * a) / y_45_scale));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.3%

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

3. Simplified 24.6%

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

4. Taylor expanded in angle around 0 53.1%

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

   1. associate-*r/ 53.1%

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

   2. *-commutative 53.1%

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

   3. unpow2 53.1%

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

   4. unpow2 53.1%

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

   5. unpow2 53.1%

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

   6. unpow2 53.1%

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

   7. unswap-sqr 62.5%

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

6. Simplified 62.5%

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

   1. times-frac 65.8%

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

   2. unswap-sqr 82.0%

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

8. Applied egg-rr 82.0%

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

   1. times-frac 87.7%

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

10. Applied egg-rr 87.7%

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

11. Final simplification 87.7%

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

Alternative 5: 35.3% accurate, 2485.0× 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 28.3%

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

3. Simplified 27.5%

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

4. Taylor expanded in b around 0 30.1%

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

   1. distribute-rgt-out 30.1%

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

   2. metadata-eval 30.1%

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

   3. mul0-rgt 37.2%

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

6. Simplified 37.2%

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

7. Final simplification 37.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023275 
(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))))