Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.5% → 92.5%

Time: 30.8s

Alternatives: 5

Speedup: 1693.0×

• 33.4% 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: 25.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.5% accurate, 76.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 6 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{a\_m}{\frac{x-scale}{\frac{b}{y-scale}}}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a\_m \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{y-scale} \cdot \left(\frac{a\_m \cdot -4}{x-scale} \cdot \left(b \cdot \frac{\frac{a\_m}{x-scale}}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 6e+183)
   (/
    (/ a_m (/ x-scale (/ b y-scale)))
    (* (/ y-scale b) (/ x-scale (* a_m -4.0))))
   (*
    (/ b y-scale)
    (* (/ (* a_m -4.0) x-scale) (* b (/ (/ a_m x-scale) y-scale))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 6e+183) {
		tmp = (a_m / (x_45_scale / (b / y_45_scale))) / ((y_45_scale / b) * (x_45_scale / (a_m * -4.0)));
	} else {
		tmp = (b / y_45_scale) * (((a_m * -4.0) / x_45_scale) * (b * ((a_m / x_45_scale) / y_45_scale)));
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    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 (a_m <= 6d+183) then
        tmp = (a_m / (x_45scale / (b / y_45scale))) / ((y_45scale / b) * (x_45scale / (a_m * (-4.0d0))))
    else
        tmp = (b / y_45scale) * (((a_m * (-4.0d0)) / x_45scale) * (b * ((a_m / x_45scale) / y_45scale)))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 6e+183) {
		tmp = (a_m / (x_45_scale / (b / y_45_scale))) / ((y_45_scale / b) * (x_45_scale / (a_m * -4.0)));
	} else {
		tmp = (b / y_45_scale) * (((a_m * -4.0) / x_45_scale) * (b * ((a_m / x_45_scale) / y_45_scale)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 6e+183:
		tmp = (a_m / (x_45_scale / (b / y_45_scale))) / ((y_45_scale / b) * (x_45_scale / (a_m * -4.0)))
	else:
		tmp = (b / y_45_scale) * (((a_m * -4.0) / x_45_scale) * (b * ((a_m / x_45_scale) / y_45_scale)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 6e+183)
		tmp = Float64(Float64(a_m / Float64(x_45_scale / Float64(b / y_45_scale))) / Float64(Float64(y_45_scale / b) * Float64(x_45_scale / Float64(a_m * -4.0))));
	else
		tmp = Float64(Float64(b / y_45_scale) * Float64(Float64(Float64(a_m * -4.0) / x_45_scale) * Float64(b * Float64(Float64(a_m / x_45_scale) / y_45_scale))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 6e+183)
		tmp = (a_m / (x_45_scale / (b / y_45_scale))) / ((y_45_scale / b) * (x_45_scale / (a_m * -4.0)));
	else
		tmp = (b / y_45_scale) * (((a_m * -4.0) / x_45_scale) * (b * ((a_m / x_45_scale) / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 6e+183], N[(N[(a$95$m / N[(x$45$scale / N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale / b), $MachinePrecision] * N[(x$45$scale / N[(a$95$m * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / y$45$scale), $MachinePrecision] * N[(N[(N[(a$95$m * -4.0), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(b * N[(N[(a$95$m / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.99999999999999992e183

   1. Initial program 28.1%

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

   3. Simplified 25.0%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 53.6%

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

   7. Simplified 53.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 70.8%

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

   9. Applied egg-rr 70.8%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 81.0%

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

   11. Applied egg-rr 81.0%

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

       1. associate-*r/ N/A

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

       2. clear-num N/A

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

       3. frac-times N/A

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

       4. *-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. associate-/l/ N/A

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

       7. associate-*r/ N/A

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

       8. *-commutative N/A

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

       9. clear-num N/A

          \[\leadsto \frac{\frac{a \cdot b}{x-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{-4 \cdot a}}}{\frac{y-scale}{b}} \]

       10. un-div-inv N/A

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

       11. associate-/l/ N/A

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

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

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

   13. Applied egg-rr 91.3%

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

   if 5.99999999999999992e183 < a

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 37.8%

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

   7. Simplified 37.8%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 59.2%

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

   9. Applied egg-rr 59.2%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

       8. *-lowering-*.f64 80.5%

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

   11. Applied egg-rr 80.5%

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

       1. times-frac N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. associate-/r/ N/A

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

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

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

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

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

       7. /-lowering-/.f64 92.6%

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

   13. Applied egg-rr 92.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{a}{\frac{x-scale}{\frac{b}{y-scale}}}}{\frac{y-scale}{b} \cdot \frac{x-scale}{a \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{y-scale} \cdot \left(\frac{a \cdot -4}{x-scale} \cdot \left(b \cdot \frac{\frac{a}{x-scale}}{y-scale}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.4%

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

3. Simplified 22.7%

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

5. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

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

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

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

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

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

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 52.1%

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

7. Simplified 52.1%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

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

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   13. /-lowering-/.f64 69.7%

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

9. Applied egg-rr 69.7%

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

    1. associate-*r* N/A

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

    2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

    8. *-lowering-*.f64 80.9%

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

11. Applied egg-rr 80.9%

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

    1. times-frac N/A

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

    2. clear-num N/A

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

    3. un-div-inv N/A

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

    4. associate-/r/ N/A

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

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

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

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

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

    7. /-lowering-/.f64 88.9%

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

13. Applied egg-rr 88.9%

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

14. Final simplification 88.9%

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

Alternative 3: 86.7% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.4%

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

3. Simplified 22.7%

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

5. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

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

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

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

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

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

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 52.1%

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

7. Simplified 52.1%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

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

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   13. /-lowering-/.f64 69.7%

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

9. Applied egg-rr 69.7%

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

    1. associate-*r* N/A

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

    2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

    8. *-lowering-*.f64 80.9%

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

11. Applied egg-rr 80.9%

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

    1. *-commutative N/A

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

    2. times-frac N/A

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

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

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

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

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

    5. /-lowering-/.f64 84.8%

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

13. Applied egg-rr 84.8%

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

14. Final simplification 84.8%

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

Alternative 4: 73.3% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.4%

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

3. Simplified 22.7%

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

5. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

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

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

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

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

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

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

   8. unpow2 N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 52.1%

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

7. Simplified 52.1%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

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

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   13. /-lowering-/.f64 69.7%

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

9. Applied egg-rr 69.7%

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

    1. associate-*r* N/A

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

    2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

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

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

    10. *-lowering-*.f64 68.3%

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

11. Applied egg-rr 68.3%

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

12. Final simplification 68.3%

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

Alternative 5: 36.4% accurate, 1693.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 0 \end{array} \]

FPCore

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

C

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

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    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

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 25.4%

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

3. Simplified 22.7%

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

5. Taylor expanded in b around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + 4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
6. Step-by-step derivation

   1. distribute-rgt-out N/A

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{\left(-4 + 4\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot 0 \]

   3. mul0-rgt 33.6%

      \[\leadsto 0 \]

7. Simplified 33.6%

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

Reproduce

herbie shell --seed 2024164 
(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))))