Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.3% → 84.6%

Time: 30.0s

Alternatives: 6

Speedup: 1693.0×

• 33.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.6% accurate, 58.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 9.7 \cdot 10^{-198}:\\ \;\;\;\;\left(\frac{a\_m \cdot \frac{a\_m \cdot \frac{-4}{x-scale}}{y-scale}}{y-scale} \cdot b\right) \cdot \frac{b}{x-scale}\\ \mathbf{elif}\;a\_m \leq 3.45 \cdot 10^{+134}:\\ \;\;\;\;a\_m \cdot \left(\frac{a\_m}{\frac{\frac{x-scale \cdot y-scale}{b}}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{x-scale} \cdot \left(b \cdot \left(a\_m \cdot \frac{\frac{4}{x-scale \cdot y-scale} \cdot \left(0 - a\_m\right)}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= a_m 9.7e-198)
   (*
    (* (/ (* a_m (/ (* a_m (/ -4.0 x-scale)) y-scale)) y-scale) b)
    (/ b x-scale))
   (if (<= a_m 3.45e+134)
     (*
      a_m
      (* (/ a_m (/ (/ (* x-scale y-scale) b) b)) (/ -4.0 (* x-scale y-scale))))
     (*
      (/ b x-scale)
      (* b (* a_m (/ (* (/ 4.0 (* x-scale y-scale)) (- 0.0 a_m)) y-scale)))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 9.7e-198) {
		tmp = (((a_m * ((a_m * (-4.0 / x_45_scale)) / y_45_scale)) / y_45_scale) * b) * (b / x_45_scale);
	} else if (a_m <= 3.45e+134) {
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	} else {
		tmp = (b / x_45_scale) * (b * (a_m * (((4.0 / (x_45_scale * y_45_scale)) * (0.0 - a_m)) / y_45_scale)));
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (a_m <= 9.7d-198) then
        tmp = (((a_m * ((a_m * ((-4.0d0) / x_45scale)) / y_45scale)) / y_45scale) * b) * (b / x_45scale)
    else if (a_m <= 3.45d+134) then
        tmp = a_m * ((a_m / (((x_45scale * y_45scale) / b) / b)) * ((-4.0d0) / (x_45scale * y_45scale)))
    else
        tmp = (b / x_45scale) * (b * (a_m * (((4.0d0 / (x_45scale * y_45scale)) * (0.0d0 - a_m)) / y_45scale)))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a_m <= 9.7e-198) {
		tmp = (((a_m * ((a_m * (-4.0 / x_45_scale)) / y_45_scale)) / y_45_scale) * b) * (b / x_45_scale);
	} else if (a_m <= 3.45e+134) {
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	} else {
		tmp = (b / x_45_scale) * (b * (a_m * (((4.0 / (x_45_scale * y_45_scale)) * (0.0 - a_m)) / y_45_scale)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a_m <= 9.7e-198:
		tmp = (((a_m * ((a_m * (-4.0 / x_45_scale)) / y_45_scale)) / y_45_scale) * b) * (b / x_45_scale)
	elif a_m <= 3.45e+134:
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)))
	else:
		tmp = (b / x_45_scale) * (b * (a_m * (((4.0 / (x_45_scale * y_45_scale)) * (0.0 - a_m)) / y_45_scale)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a_m <= 9.7e-198)
		tmp = Float64(Float64(Float64(Float64(a_m * Float64(Float64(a_m * Float64(-4.0 / x_45_scale)) / y_45_scale)) / y_45_scale) * b) * Float64(b / x_45_scale));
	elseif (a_m <= 3.45e+134)
		tmp = Float64(a_m * Float64(Float64(a_m / Float64(Float64(Float64(x_45_scale * y_45_scale) / b) / b)) * Float64(-4.0 / Float64(x_45_scale * y_45_scale))));
	else
		tmp = Float64(Float64(b / x_45_scale) * Float64(b * Float64(a_m * Float64(Float64(Float64(4.0 / Float64(x_45_scale * y_45_scale)) * Float64(0.0 - a_m)) / y_45_scale))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a_m <= 9.7e-198)
		tmp = (((a_m * ((a_m * (-4.0 / x_45_scale)) / y_45_scale)) / y_45_scale) * b) * (b / x_45_scale);
	elseif (a_m <= 3.45e+134)
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	else
		tmp = (b / x_45_scale) * (b * (a_m * (((4.0 / (x_45_scale * y_45_scale)) * (0.0 - a_m)) / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a$95$m, 9.7e-198], N[(N[(N[(N[(a$95$m * N[(N[(a$95$m * N[(-4.0 / x$45$scale), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] * b), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[a$95$m, 3.45e+134], N[(a$95$m * N[(N[(a$95$m / N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / x$45$scale), $MachinePrecision] * N[(b * N[(a$95$m * N[(N[(N[(4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(0.0 - a$95$m), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 9.70000000000000067e-198

   1. Initial program 28.0%

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

   3. Simplified 26.1%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 56.7%

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

   7. Simplified 56.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 63.5%

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

   9. Applied egg-rr 63.5%

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

       1. associate-/r/ N/A

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

       2. associate-*r* N/A

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

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

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

       4. associate-/r* N/A

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

       5. associate-/r* N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/r* N/A

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

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

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

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

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

       13. /-lowering-/.f64 80.3%

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

   11. Applied egg-rr 80.3%

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-*r/ N/A

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

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

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

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

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

       6. associate-*l/ N/A

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

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

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

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

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

       9. /-lowering-/.f64 86.3%

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

   13. Applied egg-rr 86.3%

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

   if 9.70000000000000067e-198 < a < 3.4500000000000001e134

   1. Initial program 30.4%

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

   3. Simplified 28.4%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 63.0%

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

   7. Simplified 63.0%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 74.1%

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

   9. Applied egg-rr 74.1%

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

       1. associate-*l* N/A

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

       2. associate-/l* N/A

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

       3. div-inv N/A

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

       4. clear-num N/A

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

       5. associate-*l* N/A

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. associate-/r/ N/A

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

       9. times-frac N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-/l/ N/A

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

       13. associate-/r* N/A

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

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

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

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

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

       16. /-lowering-/.f64 N/A

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

       17. associate-/l/ N/A

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

   11. Applied egg-rr 70.9%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. associate-/r* N/A

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. associate-/l/ N/A

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 84.3%

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

   13. Applied egg-rr 84.3%

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

   if 3.4500000000000001e134 < a

   1. Initial program 0.3%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 38.6%

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

   7. Simplified 38.6%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 54.2%

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

   9. Applied egg-rr 54.2%

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

       1. associate-/r/ N/A

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

       2. associate-*r* N/A

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

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

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

       4. associate-/r* N/A

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

       5. associate-/r* N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/r* N/A

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

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

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

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

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

       13. /-lowering-/.f64 81.3%

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

   11. Applied egg-rr 81.3%

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

       1. frac-2neg N/A

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

       2. associate-*l/ N/A

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

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

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

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

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

       5. associate-/l/ N/A

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

       6. distribute-neg-frac N/A

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

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

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

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

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

       11. neg-sub0 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(4, \mathsf{*.f64}\left(x-scale, y-scale\right)\right), a\right), \left(0 - y-scale\right)\right), a\right), b\right), \mathsf{/.f64}\left(b, x-scale\right)\right) \]

       12. --lowering--.f64 88.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(4, \mathsf{*.f64}\left(x-scale, y-scale\right)\right), a\right), \mathsf{\_.f64}\left(0, y-scale\right)\right), a\right), b\right), \mathsf{/.f64}\left(b, x-scale\right)\right) \]

   13. Applied egg-rr 88.7%

       \[\leadsto \left(\left(\color{blue}{\frac{\frac{4}{x-scale \cdot y-scale} \cdot a}{0 - y-scale}} \cdot a\right) \cdot b\right) \cdot \frac{b}{x-scale} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.7 \cdot 10^{-198}:\\ \;\;\;\;\left(\frac{a \cdot \frac{a \cdot \frac{-4}{x-scale}}{y-scale}}{y-scale} \cdot b\right) \cdot \frac{b}{x-scale}\\ \mathbf{elif}\;a \leq 3.45 \cdot 10^{+134}:\\ \;\;\;\;a \cdot \left(\frac{a}{\frac{\frac{x-scale \cdot y-scale}{b}}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{x-scale} \cdot \left(b \cdot \left(a \cdot \frac{\frac{4}{x-scale \cdot y-scale} \cdot \left(0 - a\right)}{y-scale}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 62.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{-4}{x-scale}}{y-scale} \cdot \left(\frac{a\_m}{\frac{y-scale}{b}} \cdot \frac{a\_m}{\frac{x-scale}{b}}\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+250}:\\ \;\;\;\;a\_m \cdot \left(\frac{a\_m}{\frac{\frac{x-scale \cdot y-scale}{b}}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot \left(a\_m \cdot \left(b \cdot \frac{a\_m}{x-scale}\right)\right)\right) \cdot \frac{-4}{y-scale}}{x-scale \cdot y-scale}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= b 1.3e-196)
   (*
    (/ (/ -4.0 x-scale) y-scale)
    (* (/ a_m (/ y-scale b)) (/ a_m (/ x-scale b))))
   (if (<= b 3.2e+250)
     (*
      a_m
      (* (/ a_m (/ (/ (* x-scale y-scale) b) b)) (/ -4.0 (* x-scale y-scale))))
     (/
      (* (* b (* a_m (* b (/ a_m x-scale)))) (/ -4.0 y-scale))
      (* x-scale y-scale)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 1.3e-196) {
		tmp = ((-4.0 / x_45_scale) / y_45_scale) * ((a_m / (y_45_scale / b)) * (a_m / (x_45_scale / b)));
	} else if (b <= 3.2e+250) {
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	} else {
		tmp = ((b * (a_m * (b * (a_m / x_45_scale)))) * (-4.0 / y_45_scale)) / (x_45_scale * y_45_scale);
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 1.3d-196) then
        tmp = (((-4.0d0) / x_45scale) / y_45scale) * ((a_m / (y_45scale / b)) * (a_m / (x_45scale / b)))
    else if (b <= 3.2d+250) then
        tmp = a_m * ((a_m / (((x_45scale * y_45scale) / b) / b)) * ((-4.0d0) / (x_45scale * y_45scale)))
    else
        tmp = ((b * (a_m * (b * (a_m / x_45scale)))) * ((-4.0d0) / y_45scale)) / (x_45scale * y_45scale)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 1.3e-196) {
		tmp = ((-4.0 / x_45_scale) / y_45_scale) * ((a_m / (y_45_scale / b)) * (a_m / (x_45_scale / b)));
	} else if (b <= 3.2e+250) {
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	} else {
		tmp = ((b * (a_m * (b * (a_m / x_45_scale)))) * (-4.0 / y_45_scale)) / (x_45_scale * y_45_scale);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 1.3e-196:
		tmp = ((-4.0 / x_45_scale) / y_45_scale) * ((a_m / (y_45_scale / b)) * (a_m / (x_45_scale / b)))
	elif b <= 3.2e+250:
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)))
	else:
		tmp = ((b * (a_m * (b * (a_m / x_45_scale)))) * (-4.0 / y_45_scale)) / (x_45_scale * y_45_scale)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 1.3e-196)
		tmp = Float64(Float64(Float64(-4.0 / x_45_scale) / y_45_scale) * Float64(Float64(a_m / Float64(y_45_scale / b)) * Float64(a_m / Float64(x_45_scale / b))));
	elseif (b <= 3.2e+250)
		tmp = Float64(a_m * Float64(Float64(a_m / Float64(Float64(Float64(x_45_scale * y_45_scale) / b) / b)) * Float64(-4.0 / Float64(x_45_scale * y_45_scale))));
	else
		tmp = Float64(Float64(Float64(b * Float64(a_m * Float64(b * Float64(a_m / x_45_scale)))) * Float64(-4.0 / y_45_scale)) / Float64(x_45_scale * y_45_scale));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 1.3e-196)
		tmp = ((-4.0 / x_45_scale) / y_45_scale) * ((a_m / (y_45_scale / b)) * (a_m / (x_45_scale / b)));
	elseif (b <= 3.2e+250)
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	else
		tmp = ((b * (a_m * (b * (a_m / x_45_scale)))) * (-4.0 / y_45_scale)) / (x_45_scale * y_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 1.3e-196], N[(N[(N[(-4.0 / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(a$95$m / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision] * N[(a$95$m / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e+250], N[(a$95$m * N[(N[(a$95$m / N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * N[(a$95$m * N[(b * N[(a$95$m / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.2999999999999999e-196

   1. Initial program 27.3%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 49.3%

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

   7. Simplified 49.3%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 59.1%

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

   9. Applied egg-rr 59.1%

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

       1. associate-*l* N/A

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

       2. associate-/l* N/A

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

       3. div-inv N/A

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

       4. clear-num N/A

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

       5. associate-*l* N/A

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. associate-/r/ N/A

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

       9. times-frac N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-/l/ N/A

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

       13. associate-/r* N/A

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

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

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

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

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

       16. /-lowering-/.f64 N/A

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

       17. associate-/l/ N/A

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

   11. Applied egg-rr 60.3%

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

       1. times-frac N/A

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

       2. times-frac N/A

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 80.6%

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

   13. Applied egg-rr 80.6%

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

   if 1.2999999999999999e-196 < b < 3.1999999999999997e250

   1. Initial program 27.1%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 74.7%

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

   7. Simplified 74.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 80.7%

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

   9. Applied egg-rr 80.7%

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

       1. associate-*l* N/A

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

       2. associate-/l* N/A

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

       3. div-inv N/A

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

       4. clear-num N/A

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

       5. associate-*l* N/A

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. associate-/r/ N/A

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

       9. times-frac N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-/l/ N/A

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

       13. associate-/r* N/A

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

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

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

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

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

       16. /-lowering-/.f64 N/A

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

       17. associate-/l/ N/A

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

   11. Applied egg-rr 85.5%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. associate-/r* N/A

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. associate-/l/ N/A

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 96.4%

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

   13. Applied egg-rr 96.4%

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

   if 3.1999999999999997e250 < b

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 30.8%

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

   7. Simplified 30.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-/r* N/A

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

      4. frac-times N/A

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

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

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

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

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 31.4%

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

   9. Applied egg-rr 31.4%

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

       1. associate-/r/ N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

       6. *-lowering-*.f64 47.3%

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

   11. Applied egg-rr 47.3%

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

       1. associate-/l* N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. /-lowering-/.f64 69.5%

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

   13. Applied egg-rr 69.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{-4}{x-scale}}{y-scale} \cdot \left(\frac{a}{\frac{y-scale}{b}} \cdot \frac{a}{\frac{x-scale}{b}}\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+250}:\\ \;\;\;\;a \cdot \left(\frac{a}{\frac{\frac{x-scale \cdot y-scale}{b}}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot \left(a \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\right) \cdot \frac{-4}{y-scale}}{x-scale \cdot y-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 62.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := a\_m \cdot \left(\frac{a\_m}{\frac{\frac{x-scale \cdot y-scale}{b}}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right)\\ \mathbf{if}\;y-scale \leq 1.55 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y-scale \leq 7.5 \cdot 10^{+155}:\\ \;\;\;\;\left(\frac{b}{x-scale} \cdot \frac{a\_m}{y-scale \cdot \frac{y-scale}{\frac{-4}{x-scale}}}\right) \cdot \left(a\_m \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          a_m
          (*
           (/ a_m (/ (/ (* x-scale y-scale) b) b))
           (/ -4.0 (* x-scale y-scale))))))
   (if (<= y-scale 1.55e+27)
     t_0
     (if (<= y-scale 7.5e+155)
       (*
        (* (/ b x-scale) (/ a_m (* y-scale (/ y-scale (/ -4.0 x-scale)))))
        (* a_m b))
       t_0))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	double tmp;
	if (y_45_scale <= 1.55e+27) {
		tmp = t_0;
	} else if (y_45_scale <= 7.5e+155) {
		tmp = ((b / x_45_scale) * (a_m / (y_45_scale * (y_45_scale / (-4.0 / x_45_scale))))) * (a_m * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * ((a_m / (((x_45scale * y_45scale) / b) / b)) * ((-4.0d0) / (x_45scale * y_45scale)))
    if (y_45scale <= 1.55d+27) then
        tmp = t_0
    else if (y_45scale <= 7.5d+155) then
        tmp = ((b / x_45scale) * (a_m / (y_45scale * (y_45scale / ((-4.0d0) / x_45scale))))) * (a_m * b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	double tmp;
	if (y_45_scale <= 1.55e+27) {
		tmp = t_0;
	} else if (y_45_scale <= 7.5e+155) {
		tmp = ((b / x_45_scale) * (a_m / (y_45_scale * (y_45_scale / (-4.0 / x_45_scale))))) * (a_m * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)))
	tmp = 0
	if y_45_scale <= 1.55e+27:
		tmp = t_0
	elif y_45_scale <= 7.5e+155:
		tmp = ((b / x_45_scale) * (a_m / (y_45_scale * (y_45_scale / (-4.0 / x_45_scale))))) * (a_m * b)
	else:
		tmp = t_0
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a_m * Float64(Float64(a_m / Float64(Float64(Float64(x_45_scale * y_45_scale) / b) / b)) * Float64(-4.0 / Float64(x_45_scale * y_45_scale))))
	tmp = 0.0
	if (y_45_scale <= 1.55e+27)
		tmp = t_0;
	elseif (y_45_scale <= 7.5e+155)
		tmp = Float64(Float64(Float64(b / x_45_scale) * Float64(a_m / Float64(y_45_scale * Float64(y_45_scale / Float64(-4.0 / x_45_scale))))) * Float64(a_m * b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	tmp = 0.0;
	if (y_45_scale <= 1.55e+27)
		tmp = t_0;
	elseif (y_45_scale <= 7.5e+155)
		tmp = ((b / x_45_scale) * (a_m / (y_45_scale * (y_45_scale / (-4.0 / x_45_scale))))) * (a_m * b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a$95$m * N[(N[(a$95$m / N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 1.55e+27], t$95$0, If[LessEqual[y$45$scale, 7.5e+155], N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(a$95$m / N[(y$45$scale * N[(y$45$scale / N[(-4.0 / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.54999999999999998e27 or 7.4999999999999999e155 < y-scale

   1. Initial program 25.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 55.5%

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

   7. Simplified 55.5%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 64.0%

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

   9. Applied egg-rr 64.0%

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

       1. associate-*l* N/A

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

       2. associate-/l* N/A

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

       3. div-inv N/A

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

       4. clear-num N/A

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

       5. associate-*l* N/A

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. associate-/r/ N/A

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

       9. times-frac N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-/l/ N/A

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

       13. associate-/r* N/A

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

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

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

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

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

       16. /-lowering-/.f64 N/A

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

       17. associate-/l/ N/A

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

   11. Applied egg-rr 67.1%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. associate-/r* N/A

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

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

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

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

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

       10. *-commutative N/A

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

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

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

       12. associate-/l/ N/A

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

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

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 85.5%

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

   13. Applied egg-rr 85.5%

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

   if 1.54999999999999998e27 < y-scale < 7.4999999999999999e155

   1. Initial program 29.5%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 65.1%

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

   7. Simplified 65.1%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 76.3%

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

   9. Applied egg-rr 76.3%

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

       1. associate-/r/ N/A

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

       2. associate-*r* N/A

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

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

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

       4. associate-/r* N/A

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

       5. associate-/r* N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/r* N/A

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

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

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

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

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

       13. /-lowering-/.f64 89.6%

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

   11. Applied egg-rr 89.6%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. div-inv N/A

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

       5. associate-/l/ N/A

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

       6. associate-*l/ N/A

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

       7. div-inv N/A

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

       8. associate-/r* N/A

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

       9. associate-/r* N/A

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

       10. associate-/r/ N/A

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

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

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

   13. Applied egg-rr 99.8%

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

Alternative 4: 84.7% accurate, 76.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{-4}{x-scale}}{y-scale} \cdot \left(\frac{a\_m}{\frac{y-scale}{b}} \cdot \frac{a\_m}{\frac{x-scale}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(\frac{a\_m}{\frac{\frac{x-scale \cdot y-scale}{b}}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= b 2.3e-197)
   (*
    (/ (/ -4.0 x-scale) y-scale)
    (* (/ a_m (/ y-scale b)) (/ a_m (/ x-scale b))))
   (*
    a_m
    (* (/ a_m (/ (/ (* x-scale y-scale) b) b)) (/ -4.0 (* x-scale y-scale))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 2.3e-197) {
		tmp = ((-4.0 / x_45_scale) / y_45_scale) * ((a_m / (y_45_scale / b)) * (a_m / (x_45_scale / b)));
	} else {
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 2.3d-197) then
        tmp = (((-4.0d0) / x_45scale) / y_45scale) * ((a_m / (y_45scale / b)) * (a_m / (x_45scale / b)))
    else
        tmp = a_m * ((a_m / (((x_45scale * y_45scale) / b) / b)) * ((-4.0d0) / (x_45scale * y_45scale)))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 2.3e-197) {
		tmp = ((-4.0 / x_45_scale) / y_45_scale) * ((a_m / (y_45_scale / b)) * (a_m / (x_45_scale / b)));
	} else {
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 2.3e-197:
		tmp = ((-4.0 / x_45_scale) / y_45_scale) * ((a_m / (y_45_scale / b)) * (a_m / (x_45_scale / b)))
	else:
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 2.3e-197)
		tmp = Float64(Float64(Float64(-4.0 / x_45_scale) / y_45_scale) * Float64(Float64(a_m / Float64(y_45_scale / b)) * Float64(a_m / Float64(x_45_scale / b))));
	else
		tmp = Float64(a_m * Float64(Float64(a_m / Float64(Float64(Float64(x_45_scale * y_45_scale) / b) / b)) * Float64(-4.0 / Float64(x_45_scale * y_45_scale))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 2.3e-197)
		tmp = ((-4.0 / x_45_scale) / y_45_scale) * ((a_m / (y_45_scale / b)) * (a_m / (x_45_scale / b)));
	else
		tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 2.3e-197], N[(N[(N[(-4.0 / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(N[(a$95$m / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision] * N[(a$95$m / N[(x$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(N[(a$95$m / N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.3000000000000001e-197

   1. Initial program 27.3%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 49.3%

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

   7. Simplified 49.3%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. /-lowering-/.f64 59.1%

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

   9. Applied egg-rr 59.1%

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

       1. associate-*l* N/A

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

       2. associate-/l* N/A

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

       3. div-inv N/A

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

       4. clear-num N/A

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

       5. associate-*l* N/A

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

       6. associate-/r* N/A

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

       7. associate-*r/ N/A

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

       8. associate-/r/ N/A

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

       9. times-frac N/A

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

       10. *-commutative N/A

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

       11. times-frac N/A

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

       12. associate-/l/ N/A

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

       13. associate-/r* N/A

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

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

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

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

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

       16. /-lowering-/.f64 N/A

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

       17. associate-/l/ N/A

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

   11. Applied egg-rr 60.3%

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

       1. times-frac N/A

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

       2. times-frac N/A

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

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

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

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

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

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

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

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

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

       7. /-lowering-/.f64 80.6%

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

   13. Applied egg-rr 80.6%

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

   if 2.3000000000000001e-197 < b

   1. Initial program 23.4%

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

   3. Simplified 22.3%

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

   5. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 68.7%

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

   7. Simplified 68.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(a \cdot a\right)\right)\right), b\right), \left(\frac{b}{x-scale}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(a, a\right)\right)\right), b\right), \left(\frac{b}{x-scale}\right)\right) \]

      13. /-lowering-/.f64 76.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(a, a\right)\right)\right), b\right), \mathsf{/.f64}\left(b, \color{blue}{x-scale}\right)\right) \]

   9. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{a \cdot a}} \cdot b\right) \cdot \frac{b}{x-scale}} \]
   10. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{a \cdot a}} \cdot \color{blue}{\left(b \cdot \frac{b}{x-scale}\right)} \]

       2. associate-/l* N/A

          \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{a \cdot a}} \cdot \frac{b \cdot b}{\color{blue}{x-scale}} \]

       3. div-inv N/A

          \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{a \cdot a}}\right) \cdot \frac{\color{blue}{b \cdot b}}{x-scale} \]

       4. clear-num N/A

          \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale}\right) \cdot \frac{b \cdot \color{blue}{b}}{x-scale} \]

       5. associate-*l* N/A

          \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \color{blue}{\left(\frac{a \cdot a}{x-scale} \cdot \frac{b \cdot b}{x-scale}\right)} \]

       6. associate-/r* N/A

          \[\leadsto \frac{\frac{-4}{y-scale}}{y-scale} \cdot \left(\color{blue}{\frac{a \cdot a}{x-scale}} \cdot \frac{b \cdot b}{x-scale}\right) \]

       7. associate-*r/ N/A

          \[\leadsto \frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{\frac{a \cdot a}{x-scale} \cdot \left(b \cdot b\right)}{\color{blue}{x-scale}} \]

       8. associate-/r/ N/A

          \[\leadsto \frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{x-scale} \]

       9. times-frac N/A

          \[\leadsto \frac{\frac{-4}{y-scale} \cdot \frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{\color{blue}{y-scale \cdot x-scale}} \]

       10. *-commutative N/A

           \[\leadsto \frac{\frac{-4}{y-scale} \cdot \frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{x-scale \cdot \color{blue}{y-scale}} \]

       11. times-frac N/A

           \[\leadsto \frac{\frac{-4}{y-scale}}{x-scale} \cdot \color{blue}{\frac{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{y-scale}} \]

       12. associate-/l/ N/A

           \[\leadsto \frac{-4}{x-scale \cdot y-scale} \cdot \frac{\color{blue}{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}}{y-scale} \]

       13. associate-/r* N/A

           \[\leadsto \frac{\frac{-4}{x-scale}}{y-scale} \cdot \frac{\color{blue}{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}}{y-scale} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-4}{x-scale}}{y-scale}\right), \color{blue}{\left(\frac{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{y-scale}\right)}\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{x-scale}\right), y-scale\right), \left(\frac{\color{blue}{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}}{y-scale}\right)\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \left(\frac{\frac{\color{blue}{a \cdot a}}{\frac{x-scale}{b \cdot b}}}{y-scale}\right)\right) \]

       17. associate-/l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot \frac{x-scale}{b \cdot b}}}\right)\right) \]

   11. Applied egg-rr 78.1%

       \[\leadsto \color{blue}{\frac{\frac{-4}{x-scale}}{y-scale} \cdot \frac{a \cdot a}{\frac{y-scale \cdot x-scale}{b \cdot b}}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{a \cdot a}{\frac{y-scale \cdot x-scale}{b \cdot b}} \cdot \color{blue}{\frac{\frac{-4}{x-scale}}{y-scale}} \]

       2. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{a}{\frac{y-scale \cdot x-scale}{b \cdot b}}\right) \cdot \frac{\color{blue}{\frac{-4}{x-scale}}}{y-scale} \]

       3. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(\frac{a}{\frac{y-scale \cdot x-scale}{b \cdot b}} \cdot \frac{\frac{-4}{x-scale}}{y-scale}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{a}{\frac{y-scale \cdot x-scale}{b \cdot b}} \cdot \frac{\frac{-4}{x-scale}}{y-scale}\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{a}{\frac{y-scale \cdot x-scale}{b \cdot b}}\right), \color{blue}{\left(\frac{\frac{-4}{x-scale}}{y-scale}\right)}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \left(\frac{y-scale \cdot x-scale}{b \cdot b}\right)\right), \left(\frac{\color{blue}{\frac{-4}{x-scale}}}{y-scale}\right)\right)\right) \]

       7. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \left(\frac{\frac{y-scale \cdot x-scale}{b}}{b}\right)\right), \left(\frac{\frac{-4}{\color{blue}{x-scale}}}{y-scale}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{y-scale \cdot x-scale}{b}\right), b\right)\right), \left(\frac{\frac{-4}{\color{blue}{x-scale}}}{y-scale}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot x-scale\right), b\right), b\right)\right), \left(\frac{\frac{-4}{x-scale}}{y-scale}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), b\right), b\right)\right), \left(\frac{\frac{-4}{x-scale}}{y-scale}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \left(\frac{\frac{-4}{x-scale}}{y-scale}\right)\right)\right) \]

       12. associate-/l/ N/A

           \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \left(\frac{-4}{\color{blue}{y-scale \cdot x-scale}}\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \mathsf{/.f64}\left(-4, \color{blue}{\left(y-scale \cdot x-scale\right)}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \mathsf{/.f64}\left(-4, \left(x-scale \cdot \color{blue}{y-scale}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 90.5%

           \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(x-scale, \color{blue}{y-scale}\right)\right)\right)\right) \]

   13. Applied egg-rr 90.5%

       \[\leadsto \color{blue}{a \cdot \left(\frac{a}{\frac{\frac{x-scale \cdot y-scale}{b}}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.0% accurate, 99.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a\_m \cdot \left(\frac{a\_m}{\frac{\frac{x-scale \cdot y-scale}{b}}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (*
  a_m
  (* (/ a_m (/ (/ (* x-scale y-scale) b) b)) (/ -4.0 (* x-scale y-scale)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = a_m * ((a_m / (((x_45scale * y_45scale) / b) / b)) * ((-4.0d0) / (x_45scale * y_45scale)))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(a_m * Float64(Float64(a_m / Float64(Float64(Float64(x_45_scale * y_45_scale) / b) / b)) * Float64(-4.0 / Float64(x_45_scale * y_45_scale))))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = a_m * ((a_m / (((x_45_scale * y_45_scale) / b) / b)) * (-4.0 / (x_45_scale * y_45_scale)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(a$95$m * N[(N[(a$95$m / N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.8%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

3. Simplified 24.0%

   \[\leadsto \color{blue}{\left(\cos \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(4 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(\frac{b \cdot b - a \cdot a}{x-scale \cdot y-scale} \cdot \frac{b \cdot b - a \cdot a}{x-scale \cdot y-scale}\right)\right)\right) + \frac{\left(\left({\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\right) \cdot \frac{-4}{x-scale}\right) \cdot \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{x-scale}}{y-scale \cdot y-scale}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \frac{-4}{{y-scale}^{2}} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-4}{{y-scale}^{2}}\right), \color{blue}{\left(\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2}}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left({y-scale}^{2}\right)\right), \left(\frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot \color{blue}{{b}^{2}}}{{x-scale}^{2}}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2} \cdot {b}^{2}}{x-scale \cdot \color{blue}{x-scale}}\right)\right) \]

   9. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{{a}^{2}}{x-scale} \cdot \color{blue}{\frac{{b}^{2}}{x-scale}}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\left(\frac{{a}^{2}}{x-scale}\right), \color{blue}{\left(\frac{{b}^{2}}{x-scale}\right)}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({a}^{2}\right), x-scale\right), \left(\frac{\color{blue}{{b}^{2}}}{x-scale}\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(a \cdot a\right), x-scale\right), \left(\frac{{\color{blue}{b}}^{2}}{x-scale}\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right), \left(\frac{{\color{blue}{b}}^{2}}{x-scale}\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right), \mathsf{/.f64}\left(\left({b}^{2}\right), \color{blue}{x-scale}\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right), \mathsf{/.f64}\left(\left(b \cdot b\right), x-scale\right)\right)\right) \]

   16. *-lowering-*.f64 56.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), x-scale\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), x-scale\right)\right)\right) \]

7. Simplified 56.5%

   \[\leadsto \color{blue}{\frac{-4}{y-scale \cdot y-scale} \cdot \left(\frac{a \cdot a}{x-scale} \cdot \frac{b \cdot b}{x-scale}\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale}\right) \cdot \color{blue}{\frac{b \cdot b}{x-scale}} \]

   2. associate-/l* N/A

      \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale}\right) \cdot \left(b \cdot \color{blue}{\frac{b}{x-scale}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale}\right) \cdot b\right) \cdot \color{blue}{\frac{b}{x-scale}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale}\right) \cdot b\right), \color{blue}{\left(\frac{b}{x-scale}\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale}\right), b\right), \left(\frac{\color{blue}{b}}{x-scale}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{a \cdot a}}\right), b\right), \left(\frac{b}{x-scale}\right)\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{a \cdot a}}\right), b\right), \left(\frac{b}{x-scale}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{y-scale \cdot y-scale}\right), \left(\frac{x-scale}{a \cdot a}\right)\right), b\right), \left(\frac{b}{x-scale}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \left(y-scale \cdot y-scale\right)\right), \left(\frac{x-scale}{a \cdot a}\right)\right), b\right), \left(\frac{b}{x-scale}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \left(\frac{x-scale}{a \cdot a}\right)\right), b\right), \left(\frac{b}{x-scale}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \left(a \cdot a\right)\right)\right), b\right), \left(\frac{b}{x-scale}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(a, a\right)\right)\right), b\right), \left(\frac{b}{x-scale}\right)\right) \]

   13. /-lowering-/.f64 65.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(y-scale, y-scale\right)\right), \mathsf{/.f64}\left(x-scale, \mathsf{*.f64}\left(a, a\right)\right)\right), b\right), \mathsf{/.f64}\left(b, \color{blue}{x-scale}\right)\right) \]

9. Applied egg-rr 65.3%

   \[\leadsto \color{blue}{\left(\frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{a \cdot a}} \cdot b\right) \cdot \frac{b}{x-scale}} \]
10. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{a \cdot a}} \cdot \color{blue}{\left(b \cdot \frac{b}{x-scale}\right)} \]

    2. associate-/l* N/A

       \[\leadsto \frac{\frac{-4}{y-scale \cdot y-scale}}{\frac{x-scale}{a \cdot a}} \cdot \frac{b \cdot b}{\color{blue}{x-scale}} \]

    3. div-inv N/A

       \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{1}{\frac{x-scale}{a \cdot a}}\right) \cdot \frac{\color{blue}{b \cdot b}}{x-scale} \]

    4. clear-num N/A

       \[\leadsto \left(\frac{-4}{y-scale \cdot y-scale} \cdot \frac{a \cdot a}{x-scale}\right) \cdot \frac{b \cdot \color{blue}{b}}{x-scale} \]

    5. associate-*l* N/A

       \[\leadsto \frac{-4}{y-scale \cdot y-scale} \cdot \color{blue}{\left(\frac{a \cdot a}{x-scale} \cdot \frac{b \cdot b}{x-scale}\right)} \]

    6. associate-/r* N/A

       \[\leadsto \frac{\frac{-4}{y-scale}}{y-scale} \cdot \left(\color{blue}{\frac{a \cdot a}{x-scale}} \cdot \frac{b \cdot b}{x-scale}\right) \]

    7. associate-*r/ N/A

       \[\leadsto \frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{\frac{a \cdot a}{x-scale} \cdot \left(b \cdot b\right)}{\color{blue}{x-scale}} \]

    8. associate-/r/ N/A

       \[\leadsto \frac{\frac{-4}{y-scale}}{y-scale} \cdot \frac{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{x-scale} \]

    9. times-frac N/A

       \[\leadsto \frac{\frac{-4}{y-scale} \cdot \frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{\color{blue}{y-scale \cdot x-scale}} \]

    10. *-commutative N/A

        \[\leadsto \frac{\frac{-4}{y-scale} \cdot \frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{x-scale \cdot \color{blue}{y-scale}} \]

    11. times-frac N/A

        \[\leadsto \frac{\frac{-4}{y-scale}}{x-scale} \cdot \color{blue}{\frac{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{y-scale}} \]

    12. associate-/l/ N/A

        \[\leadsto \frac{-4}{x-scale \cdot y-scale} \cdot \frac{\color{blue}{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}}{y-scale} \]

    13. associate-/r* N/A

        \[\leadsto \frac{\frac{-4}{x-scale}}{y-scale} \cdot \frac{\color{blue}{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}}{y-scale} \]

    14. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-4}{x-scale}}{y-scale}\right), \color{blue}{\left(\frac{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}{y-scale}\right)}\right) \]

    15. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-4}{x-scale}\right), y-scale\right), \left(\frac{\color{blue}{\frac{a \cdot a}{\frac{x-scale}{b \cdot b}}}}{y-scale}\right)\right) \]

    16. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \left(\frac{\frac{\color{blue}{a \cdot a}}{\frac{x-scale}{b \cdot b}}}{y-scale}\right)\right) \]

    17. associate-/l/ N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-4, x-scale\right), y-scale\right), \left(\frac{a \cdot a}{\color{blue}{y-scale \cdot \frac{x-scale}{b \cdot b}}}\right)\right) \]

11. Applied egg-rr 66.9%

    \[\leadsto \color{blue}{\frac{\frac{-4}{x-scale}}{y-scale} \cdot \frac{a \cdot a}{\frac{y-scale \cdot x-scale}{b \cdot b}}} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{a \cdot a}{\frac{y-scale \cdot x-scale}{b \cdot b}} \cdot \color{blue}{\frac{\frac{-4}{x-scale}}{y-scale}} \]

    2. associate-/l* N/A

       \[\leadsto \left(a \cdot \frac{a}{\frac{y-scale \cdot x-scale}{b \cdot b}}\right) \cdot \frac{\color{blue}{\frac{-4}{x-scale}}}{y-scale} \]

    3. associate-*l* N/A

       \[\leadsto a \cdot \color{blue}{\left(\frac{a}{\frac{y-scale \cdot x-scale}{b \cdot b}} \cdot \frac{\frac{-4}{x-scale}}{y-scale}\right)} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{a}{\frac{y-scale \cdot x-scale}{b \cdot b}} \cdot \frac{\frac{-4}{x-scale}}{y-scale}\right)}\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(\frac{a}{\frac{y-scale \cdot x-scale}{b \cdot b}}\right), \color{blue}{\left(\frac{\frac{-4}{x-scale}}{y-scale}\right)}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \left(\frac{y-scale \cdot x-scale}{b \cdot b}\right)\right), \left(\frac{\color{blue}{\frac{-4}{x-scale}}}{y-scale}\right)\right)\right) \]

    7. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \left(\frac{\frac{y-scale \cdot x-scale}{b}}{b}\right)\right), \left(\frac{\frac{-4}{\color{blue}{x-scale}}}{y-scale}\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{y-scale \cdot x-scale}{b}\right), b\right)\right), \left(\frac{\frac{-4}{\color{blue}{x-scale}}}{y-scale}\right)\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y-scale \cdot x-scale\right), b\right), b\right)\right), \left(\frac{\frac{-4}{x-scale}}{y-scale}\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x-scale \cdot y-scale\right), b\right), b\right)\right), \left(\frac{\frac{-4}{x-scale}}{y-scale}\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \left(\frac{\frac{-4}{x-scale}}{y-scale}\right)\right)\right) \]

    12. associate-/l/ N/A

        \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \left(\frac{-4}{\color{blue}{y-scale \cdot x-scale}}\right)\right)\right) \]

    13. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \mathsf{/.f64}\left(-4, \color{blue}{\left(y-scale \cdot x-scale\right)}\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \mathsf{/.f64}\left(-4, \left(x-scale \cdot \color{blue}{y-scale}\right)\right)\right)\right) \]

    15. *-lowering-*.f64 85.6%

        \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{/.f64}\left(a, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x-scale, y-scale\right), b\right), b\right)\right), \mathsf{/.f64}\left(-4, \mathsf{*.f64}\left(x-scale, \color{blue}{y-scale}\right)\right)\right)\right) \]

13. Applied egg-rr 85.6%

    \[\leadsto \color{blue}{a \cdot \left(\frac{a}{\frac{\frac{x-scale \cdot y-scale}{b}}{b}} \cdot \frac{-4}{x-scale \cdot y-scale}\right)} \]
14. Add Preprocessing

Alternative 6: 35.2% accurate, 1693.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 0 \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale) :precision binary64 0.0)

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.0d0
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return 0.0
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 25.8%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

3. Simplified 24.0%

   \[\leadsto \color{blue}{\left(\cos \left(angle \cdot \frac{\pi}{180}\right) \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(4 \cdot \left(\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right) \cdot \left(\frac{b \cdot b - a \cdot a}{x-scale \cdot y-scale} \cdot \frac{b \cdot b - a \cdot a}{x-scale \cdot y-scale}\right)\right)\right) + \frac{\left(\left({\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\right) \cdot \frac{-4}{x-scale}\right) \cdot \frac{{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}{x-scale}}{y-scale \cdot y-scale}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + 4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
6. Step-by-step derivation

   1. distribute-rgt-out N/A

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{\left(-4 + 4\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot 0 \]

   3. mul0-rgt 34.9%

      \[\leadsto 0 \]

7. Simplified 34.9%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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))))