Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.1% → 86.5%

Time: 25.9s

Alternatives: 7

Speedup: 99.6×

• 33.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 2.8540101863161513e+289
  2. b = 5.884716861743953e-273
  3. angle = 6.3967305664829785e+100
  4. x-scale = -3.952404150311208e-218
  5. y-scale = -2.9215040564150293e-181

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: 25.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.5% accurate, 76.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{-4 \cdot a}{\frac{x-scale}{b\_m}}\\ \mathbf{if}\;b\_m \leq 3.8 \cdot 10^{-147}:\\ \;\;\;\;a \cdot \frac{t\_0}{\frac{y-scale}{\frac{\frac{b\_m}{x-scale}}{y-scale}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{\frac{b\_m}{y-scale}}{\frac{x-scale}{a}}}{y-scale}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 a) (/ x-scale b_m))))
   (if (<= b_m 3.8e-147)
     (* a (/ t_0 (/ y-scale (/ (/ b_m x-scale) y-scale))))
     (/ (* t_0 (/ (/ b_m y-scale) (/ x-scale a))) y-scale))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (-4.0 * a) / (x_45_scale / b_m);
	double tmp;
	if (b_m <= 3.8e-147) {
		tmp = a * (t_0 / (y_45_scale / ((b_m / x_45_scale) / y_45_scale)));
	} else {
		tmp = (t_0 * ((b_m / y_45_scale) / (x_45_scale / a))) / y_45_scale;
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-4.0d0) * a) / (x_45scale / b_m)
    if (b_m <= 3.8d-147) then
        tmp = a * (t_0 / (y_45scale / ((b_m / x_45scale) / y_45scale)))
    else
        tmp = (t_0 * ((b_m / y_45scale) / (x_45scale / a))) / y_45scale
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (-4.0 * a) / (x_45_scale / b_m);
	double tmp;
	if (b_m <= 3.8e-147) {
		tmp = a * (t_0 / (y_45_scale / ((b_m / x_45_scale) / y_45_scale)));
	} else {
		tmp = (t_0 * ((b_m / y_45_scale) / (x_45_scale / a))) / y_45_scale;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (-4.0 * a) / (x_45_scale / b_m)
	tmp = 0
	if b_m <= 3.8e-147:
		tmp = a * (t_0 / (y_45_scale / ((b_m / x_45_scale) / y_45_scale)))
	else:
		tmp = (t_0 * ((b_m / y_45_scale) / (x_45_scale / a))) / y_45_scale
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(-4.0 * a) / Float64(x_45_scale / b_m))
	tmp = 0.0
	if (b_m <= 3.8e-147)
		tmp = Float64(a * Float64(t_0 / Float64(y_45_scale / Float64(Float64(b_m / x_45_scale) / y_45_scale))));
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(b_m / y_45_scale) / Float64(x_45_scale / a))) / y_45_scale);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (-4.0 * a) / (x_45_scale / b_m);
	tmp = 0.0;
	if (b_m <= 3.8e-147)
		tmp = a * (t_0 / (y_45_scale / ((b_m / x_45_scale) / y_45_scale)));
	else
		tmp = (t_0 * ((b_m / y_45_scale) / (x_45_scale / a))) / y_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] / N[(x$45$scale / b$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 3.8e-147], N[(a * N[(t$95$0 / N[(y$45$scale / N[(N[(b$95$m / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(b$95$m / y$45$scale), $MachinePrecision] / N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.80000000000000028e-147

   1. Initial program 24.9%

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

   3. Simplified 18.2%

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 50.2%

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

   7. Simplified 50.2%

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

      1. frac-times N/A

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

      2. swap-sqr N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 52.6%

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

   9. Applied egg-rr 52.6%

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

       1. associate-*r/ N/A

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

       2. associate-/r* N/A

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

       3. associate-*r* N/A

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

       4. associate-/l/ N/A

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

       5. associate-/r* N/A

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

       6. associate-/l* N/A

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

       7. times-frac N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 70.1%

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

   11. Applied egg-rr 70.1%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

   13. Applied egg-rr 83.4%

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

   if 3.80000000000000028e-147 < b

   1. Initial program 19.4%

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

   3. Simplified 18.4%

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 60.5%

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

   7. Simplified 60.5%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 68.4%

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

   9. Applied egg-rr 68.4%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

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

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. *-lowering-*.f64 72.1%

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

   11. Applied egg-rr 72.1%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

       5. associate-/l* N/A

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

       8. associate-*l* N/A

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

       9. associate-/r* N/A

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

       10. associate-/r/ N/A

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

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

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

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

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

       13. /-lowering-/.f64 N/A

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

       14. /-lowering-/.f64 N/A

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

       15. clear-num N/A

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

       16. un-div-inv N/A

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

       17. /-lowering-/.f64 N/A

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

       18. *-commutative N/A

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

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

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

       20. /-lowering-/.f64 89.1%

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

   13. Applied egg-rr 89.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-147}:\\ \;\;\;\;a \cdot \frac{\frac{-4 \cdot a}{\frac{x-scale}{b}}}{\frac{y-scale}{\frac{\frac{b}{x-scale}}{y-scale}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot a}{\frac{x-scale}{b}} \cdot \frac{\frac{b}{y-scale}}{\frac{x-scale}{a}}}{y-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 76.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{\frac{b\_m}{x-scale}}{y-scale}\\ \mathbf{if}\;y-scale \leq 2.7 \cdot 10^{+216}:\\ \;\;\;\;a \cdot \frac{\frac{-4 \cdot a}{\frac{x-scale}{b\_m}}}{\frac{y-scale}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(b\_m \cdot \left(-4 \cdot a\right)\right)}{y-scale} \cdot \frac{t\_0}{x-scale}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (/ b_m x-scale) y-scale)))
   (if (<= y-scale 2.7e+216)
     (* a (/ (/ (* -4.0 a) (/ x-scale b_m)) (/ y-scale t_0)))
     (* (/ (* a (* b_m (* -4.0 a))) y-scale) (/ t_0 x-scale)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m / x_45_scale) / y_45_scale;
	double tmp;
	if (y_45_scale <= 2.7e+216) {
		tmp = a * (((-4.0 * a) / (x_45_scale / b_m)) / (y_45_scale / t_0));
	} else {
		tmp = ((a * (b_m * (-4.0 * a))) / y_45_scale) * (t_0 / x_45_scale);
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b_m / x_45scale) / y_45scale
    if (y_45scale <= 2.7d+216) then
        tmp = a * ((((-4.0d0) * a) / (x_45scale / b_m)) / (y_45scale / t_0))
    else
        tmp = ((a * (b_m * ((-4.0d0) * a))) / y_45scale) * (t_0 / x_45scale)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m / x_45_scale) / y_45_scale;
	double tmp;
	if (y_45_scale <= 2.7e+216) {
		tmp = a * (((-4.0 * a) / (x_45_scale / b_m)) / (y_45_scale / t_0));
	} else {
		tmp = ((a * (b_m * (-4.0 * a))) / y_45_scale) * (t_0 / x_45_scale);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (b_m / x_45_scale) / y_45_scale
	tmp = 0
	if y_45_scale <= 2.7e+216:
		tmp = a * (((-4.0 * a) / (x_45_scale / b_m)) / (y_45_scale / t_0))
	else:
		tmp = ((a * (b_m * (-4.0 * a))) / y_45_scale) * (t_0 / x_45_scale)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b_m / x_45_scale) / y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 2.7e+216)
		tmp = Float64(a * Float64(Float64(Float64(-4.0 * a) / Float64(x_45_scale / b_m)) / Float64(y_45_scale / t_0)));
	else
		tmp = Float64(Float64(Float64(a * Float64(b_m * Float64(-4.0 * a))) / y_45_scale) * Float64(t_0 / x_45_scale));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (b_m / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (y_45_scale <= 2.7e+216)
		tmp = a * (((-4.0 * a) / (x_45_scale / b_m)) / (y_45_scale / t_0));
	else
		tmp = ((a * (b_m * (-4.0 * a))) / y_45_scale) * (t_0 / x_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b$95$m / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[y$45$scale, 2.7e+216], N[(a * N[(N[(N[(-4.0 * a), $MachinePrecision] / N[(x$45$scale / b$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(b$95$m * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(t$95$0 / x$45$scale), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.7000000000000001e216

   1. Initial program 21.7%

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

   3. Simplified 16.8%

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 55.0%

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

   7. Simplified 55.0%

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

      1. frac-times N/A

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

      2. swap-sqr N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 59.1%

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

   9. Applied egg-rr 59.1%

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

       1. associate-*r/ N/A

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

       2. associate-/r* N/A

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

       3. associate-*r* N/A

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

       4. associate-/l/ N/A

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

       5. associate-/r* N/A

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

       6. associate-/l* N/A

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

       7. times-frac N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 74.0%

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

   11. Applied egg-rr 74.0%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

   13. Applied egg-rr 86.8%

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

   if 2.7000000000000001e216 < y-scale

   1. Initial program 35.5%

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

   3. Simplified 35.4%

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 45.8%

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

   7. Simplified 45.8%

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

      1. frac-times N/A

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

      2. swap-sqr N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 46.2%

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

   9. Applied egg-rr 46.2%

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

       1. associate-*r/ N/A

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

       2. associate-/r* N/A

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

       3. associate-*r* N/A

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

       4. associate-/l/ N/A

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

       5. associate-/r* N/A

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

       6. associate-/l* N/A

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

       7. times-frac N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 51.0%

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

   11. Applied egg-rr 51.0%

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

       1. *-commutative N/A

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

       2. associate-/l/ N/A

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

       3. associate-*l/ N/A

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

       4. associate-*r/ N/A

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

       5. associate-/l/ N/A

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

       6. associate-*r* N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

       9. associate-/r/ N/A

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

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

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

   13. Applied egg-rr 74.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.7 \cdot 10^{+216}:\\ \;\;\;\;a \cdot \frac{\frac{-4 \cdot a}{\frac{x-scale}{b}}}{\frac{y-scale}{\frac{\frac{b}{x-scale}}{y-scale}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(b \cdot \left(-4 \cdot a\right)\right)}{y-scale} \cdot \frac{\frac{\frac{b}{x-scale}}{y-scale}}{x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 76.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{\frac{b\_m}{x-scale}}{y-scale}\\ \mathbf{if}\;angle \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;\frac{a \cdot \left(b\_m \cdot \left(-4 \cdot a\right)\right)}{y-scale} \cdot \frac{t\_0}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(-4 \cdot a\right) \cdot \frac{t\_0}{\frac{x-scale}{\frac{b\_m}{y-scale}}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (/ b_m x-scale) y-scale)))
   (if (<= angle 1.25e-17)
     (* (/ (* a (* b_m (* -4.0 a))) y-scale) (/ t_0 x-scale))
     (* a (* (* -4.0 a) (/ t_0 (/ x-scale (/ b_m y-scale))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m / x_45_scale) / y_45_scale;
	double tmp;
	if (angle <= 1.25e-17) {
		tmp = ((a * (b_m * (-4.0 * a))) / y_45_scale) * (t_0 / x_45_scale);
	} else {
		tmp = a * ((-4.0 * a) * (t_0 / (x_45_scale / (b_m / y_45_scale))));
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b_m / x_45scale) / y_45scale
    if (angle <= 1.25d-17) then
        tmp = ((a * (b_m * ((-4.0d0) * a))) / y_45scale) * (t_0 / x_45scale)
    else
        tmp = a * (((-4.0d0) * a) * (t_0 / (x_45scale / (b_m / y_45scale))))
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b_m / x_45_scale) / y_45_scale;
	double tmp;
	if (angle <= 1.25e-17) {
		tmp = ((a * (b_m * (-4.0 * a))) / y_45_scale) * (t_0 / x_45_scale);
	} else {
		tmp = a * ((-4.0 * a) * (t_0 / (x_45_scale / (b_m / y_45_scale))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (b_m / x_45_scale) / y_45_scale
	tmp = 0
	if angle <= 1.25e-17:
		tmp = ((a * (b_m * (-4.0 * a))) / y_45_scale) * (t_0 / x_45_scale)
	else:
		tmp = a * ((-4.0 * a) * (t_0 / (x_45_scale / (b_m / y_45_scale))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b_m / x_45_scale) / y_45_scale)
	tmp = 0.0
	if (angle <= 1.25e-17)
		tmp = Float64(Float64(Float64(a * Float64(b_m * Float64(-4.0 * a))) / y_45_scale) * Float64(t_0 / x_45_scale));
	else
		tmp = Float64(a * Float64(Float64(-4.0 * a) * Float64(t_0 / Float64(x_45_scale / Float64(b_m / y_45_scale)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (b_m / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (angle <= 1.25e-17)
		tmp = ((a * (b_m * (-4.0 * a))) / y_45_scale) * (t_0 / x_45_scale);
	else
		tmp = a * ((-4.0 * a) * (t_0 / (x_45_scale / (b_m / y_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b$95$m / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[angle, 1.25e-17], N[(N[(N[(a * N[(b$95$m * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(t$95$0 / x$45$scale), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(-4.0 * a), $MachinePrecision] * N[(t$95$0 / N[(x$45$scale / N[(b$95$m / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.25e-17

   1. Initial program 25.7%

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

   3. Simplified 19.7%

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 54.7%

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

   7. Simplified 54.7%

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

      1. frac-times N/A

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

      2. swap-sqr N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 60.2%

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

   9. Applied egg-rr 60.2%

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

       1. associate-*r/ N/A

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

       2. associate-/r* N/A

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

       3. associate-*r* N/A

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

       4. associate-/l/ N/A

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

       5. associate-/r* N/A

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

       6. associate-/l* N/A

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

       7. times-frac N/A

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 72.4%

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

   11. Applied egg-rr 72.4%

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

       1. *-commutative N/A

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

       2. associate-/l/ N/A

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

       3. associate-*l/ N/A

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

       4. associate-*r/ N/A

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

       5. associate-/l/ N/A

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

       6. associate-*r* N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

       9. associate-/r/ N/A

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

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

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

   13. Applied egg-rr 81.5%

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

   if 1.25e-17 < angle

   1. Initial program 14.1%

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

   3. Simplified 14.1%

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. /-lowering-/.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 53.0%

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

   7. Simplified 53.0%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 63.3%

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

   9. Applied egg-rr 63.3%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. associate-/r* N/A

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

       6. times-frac N/A

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

       7. times-frac N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. associate-*r* N/A

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

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

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

   11. Applied egg-rr 64.8%

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

       1. associate-/l/ N/A

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

       2. associate-/l* N/A

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

       3. associate-/r* N/A

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

       4. associate-/l/ N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. clear-num N/A

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

       8. frac-times N/A

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

       9. associate-/l* N/A

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

       10. *-commutative N/A

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

       11. *-rgt-identity N/A

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

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

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

       13. associate-/r* N/A

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

       14. /-lowering-/.f64 N/A

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

       15. /-lowering-/.f64 N/A

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

       16. associate-/l* N/A

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

       17. clear-num N/A

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

       18. un-div-inv N/A

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

       19. /-lowering-/.f64 N/A

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

       20. /-lowering-/.f64 83.9%

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

   13. Applied egg-rr 83.9%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;\frac{a \cdot \left(b \cdot \left(-4 \cdot a\right)\right)}{y-scale} \cdot \frac{\frac{\frac{b}{x-scale}}{y-scale}}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(-4 \cdot a\right) \cdot \frac{\frac{\frac{b}{x-scale}}{y-scale}}{\frac{x-scale}{\frac{b}{y-scale}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Simplified 18.3%

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

   13. /-lowering-/.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 54.2%

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

7. Simplified 54.2%

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

   1. associate-*r* N/A

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

   2. associate-/r* N/A

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

   3. associate-*r/ N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   13. /-lowering-/.f64 63.7%

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

9. Applied egg-rr 63.7%

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

    1. associate-/l* N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. associate-/r* N/A

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

    6. times-frac N/A

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

    7. times-frac N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. associate-*r* N/A

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

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

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

11. Applied egg-rr 69.7%

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

    1. associate-/l/ N/A

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

    2. associate-/l* N/A

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

    3. associate-/r* N/A

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

    4. associate-/l/ N/A

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

    5. associate-*l/ N/A

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

    6. *-commutative N/A

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

    7. clear-num N/A

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

    8. frac-times N/A

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

    9. associate-/l* N/A

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

    10. *-commutative N/A

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

    11. *-rgt-identity N/A

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

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

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

    13. associate-/r* N/A

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

    14. /-lowering-/.f64 N/A

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

    15. /-lowering-/.f64 N/A

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

    16. associate-/l* N/A

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

    17. clear-num N/A

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

    18. un-div-inv N/A

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

    19. /-lowering-/.f64 N/A

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

    20. /-lowering-/.f64 80.7%

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

13. Applied egg-rr 80.7%

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

14. Final simplification 80.7%

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

Alternative 5: 72.7% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Simplified 18.3%

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

   13. /-lowering-/.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 54.2%

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

7. Simplified 54.2%

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

   1. frac-times N/A

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

   2. swap-sqr N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

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

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

   11. *-lowering-*.f64 58.1%

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

9. Applied egg-rr 58.1%

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

    1. associate-*r/ N/A

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

    2. associate-/r* N/A

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

    3. associate-*r* N/A

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

    4. associate-/l/ N/A

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

    5. associate-/r* N/A

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

    6. associate-/l* N/A

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

    7. times-frac N/A

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

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

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

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

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

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

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

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

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

    12. *-lowering-*.f64 72.2%

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

11. Applied egg-rr 72.2%

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

Alternative 6: 71.8% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Simplified 18.3%

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

   13. /-lowering-/.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 54.2%

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

7. Simplified 54.2%

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

   1. frac-times N/A

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

   2. swap-sqr N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

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

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

   11. *-lowering-*.f64 58.1%

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

9. Applied egg-rr 58.1%

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

    1. associate-/r* N/A

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

    2. associate-*r* N/A

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

    3. associate-/r* N/A

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

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

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

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

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

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

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

    7. *-lowering-*.f64 71.1%

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

11. Applied egg-rr 71.1%

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

Alternative 7: 60.6% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Simplified 18.3%

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

   13. /-lowering-/.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 54.2%

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

7. Simplified 54.2%

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

   1. frac-times N/A

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

   2. swap-sqr N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

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

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

   11. *-lowering-*.f64 58.1%

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

9. Applied egg-rr 58.1%

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

Reproduce

herbie shell --seed 2024191 
(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))))