Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.8% → 94.0%

Time: 1.8min

Alternatives: 4

Speedup: 1693.0×

• 33.8% 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: 94.0% accurate, 14.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2 \cdot 10^{+206}:\\ \;\;\;\;-4 \cdot {\left(a \cdot \frac{b\_m}{x-scale \cdot y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale} \cdot \frac{b\_m}{-y-scale}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2e+206)
   (* -4.0 (pow (* a (/ b_m (* x-scale y-scale))) 2.0))
   (* -4.0 (pow (* (/ a x-scale) (/ b_m (- y-scale))) 2.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2e+206) {
		tmp = -4.0 * pow((a * (b_m / (x_45_scale * y_45_scale))), 2.0);
	} else {
		tmp = -4.0 * pow(((a / x_45_scale) * (b_m / -y_45_scale)), 2.0);
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b_m <= 2d+206) then
        tmp = (-4.0d0) * ((a * (b_m / (x_45scale * y_45scale))) ** 2.0d0)
    else
        tmp = (-4.0d0) * (((a / x_45scale) * (b_m / -y_45scale)) ** 2.0d0)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2e+206) {
		tmp = -4.0 * Math.pow((a * (b_m / (x_45_scale * y_45_scale))), 2.0);
	} else {
		tmp = -4.0 * Math.pow(((a / x_45_scale) * (b_m / -y_45_scale)), 2.0);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 2e+206:
		tmp = -4.0 * math.pow((a * (b_m / (x_45_scale * y_45_scale))), 2.0)
	else:
		tmp = -4.0 * math.pow(((a / x_45_scale) * (b_m / -y_45_scale)), 2.0)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 2e+206)
		tmp = Float64(-4.0 * (Float64(a * Float64(b_m / Float64(x_45_scale * y_45_scale))) ^ 2.0));
	else
		tmp = Float64(-4.0 * (Float64(Float64(a / x_45_scale) * Float64(b_m / Float64(-y_45_scale))) ^ 2.0));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 2e+206)
		tmp = -4.0 * ((a * (b_m / (x_45_scale * y_45_scale))) ^ 2.0);
	else
		tmp = -4.0 * (((a / x_45_scale) * (b_m / -y_45_scale)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 2e+206], N[(-4.0 * N[Power[N[(a * N[(b$95$m / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[Power[N[(N[(a / x$45$scale), $MachinePrecision] * N[(b$95$m / (-y$45$scale)), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.0000000000000001e206

   1. Initial program 29.9%

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

   3. Taylor expanded in angle around 0 51.4%

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

      1. associate-/l* 52.2%

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

   5. Simplified 52.2%

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

      1. add-sqr-sqrt 52.2%

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

      2. pow2 52.2%

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

      3. associate-*r/ 51.4%

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

      4. pow-prod-down 60.8%

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

      5. pow-prod-down 79.1%

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

   7. Applied egg-rr 79.1%

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

   8. Taylor expanded in a around 0 95.0%

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

      1. associate-/l* 95.3%

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

   10. Simplified 95.3%

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

   if 2.0000000000000001e206 < b

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0 48.3%

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

      1. associate-/l* 48.3%

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

   5. Simplified 48.3%

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

      1. add-sqr-sqrt 48.3%

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

      2. pow2 48.3%

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

      3. associate-*r/ 48.3%

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

      4. pow-prod-down 64.0%

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

      5. pow-prod-down 74.2%

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

   7. Applied egg-rr 74.2%

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

   8. Taylor expanded in a around -inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. times-frac 96.4%

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

      3. distribute-rgt-neg-in 96.4%

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

   10. Simplified 96.4%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+206}:\\ \;\;\;\;-4 \cdot {\left(a \cdot \frac{b}{x-scale \cdot y-scale}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot {\left(\frac{a}{x-scale} \cdot \frac{b}{-y-scale}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 15.4× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* -4.0 (pow (* a (/ b_m (* x-scale y-scale))) 2.0)))

C

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

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * ((a * (b_m / (x_45scale * y_45scale))) ** 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(-4.0 * N[Power[N[(a * N[(b$95$m / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.6%

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

3. Taylor expanded in angle around 0 51.0%

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

   1. associate-/l* 51.8%

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

5. Simplified 51.8%

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

   1. add-sqr-sqrt 51.8%

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

   2. pow2 51.8%

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

   3. associate-*r/ 51.1%

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

   4. pow-prod-down 61.2%

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

   5. pow-prod-down 78.5%

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

7. Applied egg-rr 78.5%

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

8. Taylor expanded in a around 0 94.0%

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

   1. associate-/l* 94.6%

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

10. Simplified 94.6%

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

Alternative 3: 77.1% accurate, 80.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{1}{x-scale \cdot y-scale}\\ -4 \cdot \left(\left(\left(b\_m \cdot a\right) \cdot \left(b\_m \cdot a\right)\right) \cdot \left(t\_0 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x-scale y-scale))))
   (* -4.0 (* (* (* b_m a) (* b_m a)) (* t_0 t_0)))))

C

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

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = 1.0d0 / (x_45scale * y_45scale)
    code = (-4.0d0) * (((b_m * a) * (b_m * a)) * (t_0 * t_0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.6%

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

3. Taylor expanded in angle around 0 51.0%

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

   1. associate-/l* 51.8%

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

5. Simplified 51.8%

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

   1. *-commutative 51.8%

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

   2. div-inv 51.8%

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

   3. pow-prod-down 65.1%

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

   4. *-commutative 65.1%

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

   5. pow-flip 65.1%

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

   6. *-commutative 65.1%

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

   7. metadata-eval 65.1%

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

7. Applied egg-rr 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 65.5%

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

   3. unpow2 65.5%

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

   4. unpow2 65.5%

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

   5. swap-sqr 78.5%

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

   6. unpow2 78.5%

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

9. Simplified 78.5%

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

    1. unpow2 78.5%

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

11. Applied egg-rr 78.5%

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

    1. sqr-pow 78.5%

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

    2. metadata-eval 78.5%

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

    3. unpow-1 78.5%

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

    4. metadata-eval 78.5%

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

    5. unpow-1 78.5%

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

13. Applied egg-rr 78.5%

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

14. Final simplification 78.5%

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

Alternative 4: 35.1% accurate, 1693.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 0 \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale) :precision binary64 0.0)

C

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

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.0d0
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 26.6%

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

3. Simplified 24.1%

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

5. Taylor expanded in a around 0 24.9%

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

   1. distribute-rgt-out 24.9%

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

   2. metadata-eval 24.9%

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

   3. mul0-rgt 32.5%

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

7. Simplified 32.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))