Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.3% → 82.8%

Time: 1.3min

Alternatives: 5

Speedup: 2485.0×

• 34.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.3% 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: 82.8% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \frac{b}{y-scale}\\ t_1 := \frac{a}{x-scale} \cdot \frac{a}{x-scale}\\ t_2 := \frac{a}{\frac{y-scale}{b}}\\ t_3 := -4 \cdot \frac{t_2 \cdot t_2}{x-scale \cdot x-scale}\\ \mathbf{if}\;x-scale \leq -1.8 \cdot 10^{+181}:\\ \;\;\;\;-4 \cdot \frac{{\left(\frac{a}{x-scale}\right)}^{2} \cdot t_0}{y-scale}\\ \mathbf{elif}\;x-scale \leq -1.14 \cdot 10^{-160}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x-scale \leq 6 \cdot 10^{-160}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot \frac{t_0}{y-scale}\right)\\ \mathbf{elif}\;x-scale \leq 1.05 \cdot 10^{+122}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* b (/ b y-scale)))
        (t_1 (* (/ a x-scale) (/ a x-scale)))
        (t_2 (/ a (/ y-scale b)))
        (t_3 (* -4.0 (/ (* t_2 t_2) (* x-scale x-scale)))))
   (if (<= x-scale -1.8e+181)
     (* -4.0 (/ (* (pow (/ a x-scale) 2.0) t_0) y-scale))
     (if (<= x-scale -1.14e-160)
       t_3
       (if (<= x-scale 6e-160)
         (* -4.0 (* t_1 (/ t_0 y-scale)))
         (if (<= x-scale 1.05e+122)
           t_3
           (* -4.0 (* t_1 (* (/ b y-scale) (/ b y-scale))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * (b / y_45_scale);
	double t_1 = (a / x_45_scale) * (a / x_45_scale);
	double t_2 = a / (y_45_scale / b);
	double t_3 = -4.0 * ((t_2 * t_2) / (x_45_scale * x_45_scale));
	double tmp;
	if (x_45_scale <= -1.8e+181) {
		tmp = -4.0 * ((pow((a / x_45_scale), 2.0) * t_0) / y_45_scale);
	} else if (x_45_scale <= -1.14e-160) {
		tmp = t_3;
	} else if (x_45_scale <= 6e-160) {
		tmp = -4.0 * (t_1 * (t_0 / y_45_scale));
	} else if (x_45_scale <= 1.05e+122) {
		tmp = t_3;
	} else {
		tmp = -4.0 * (t_1 * ((b / y_45_scale) * (b / y_45_scale)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = b * (b / y_45scale)
    t_1 = (a / x_45scale) * (a / x_45scale)
    t_2 = a / (y_45scale / b)
    t_3 = (-4.0d0) * ((t_2 * t_2) / (x_45scale * x_45scale))
    if (x_45scale <= (-1.8d+181)) then
        tmp = (-4.0d0) * ((((a / x_45scale) ** 2.0d0) * t_0) / y_45scale)
    else if (x_45scale <= (-1.14d-160)) then
        tmp = t_3
    else if (x_45scale <= 6d-160) then
        tmp = (-4.0d0) * (t_1 * (t_0 / y_45scale))
    else if (x_45scale <= 1.05d+122) then
        tmp = t_3
    else
        tmp = (-4.0d0) * (t_1 * ((b / y_45scale) * (b / y_45scale)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * (b / y_45_scale);
	double t_1 = (a / x_45_scale) * (a / x_45_scale);
	double t_2 = a / (y_45_scale / b);
	double t_3 = -4.0 * ((t_2 * t_2) / (x_45_scale * x_45_scale));
	double tmp;
	if (x_45_scale <= -1.8e+181) {
		tmp = -4.0 * ((Math.pow((a / x_45_scale), 2.0) * t_0) / y_45_scale);
	} else if (x_45_scale <= -1.14e-160) {
		tmp = t_3;
	} else if (x_45_scale <= 6e-160) {
		tmp = -4.0 * (t_1 * (t_0 / y_45_scale));
	} else if (x_45_scale <= 1.05e+122) {
		tmp = t_3;
	} else {
		tmp = -4.0 * (t_1 * ((b / y_45_scale) * (b / y_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b * (b / y_45_scale)
	t_1 = (a / x_45_scale) * (a / x_45_scale)
	t_2 = a / (y_45_scale / b)
	t_3 = -4.0 * ((t_2 * t_2) / (x_45_scale * x_45_scale))
	tmp = 0
	if x_45_scale <= -1.8e+181:
		tmp = -4.0 * ((math.pow((a / x_45_scale), 2.0) * t_0) / y_45_scale)
	elif x_45_scale <= -1.14e-160:
		tmp = t_3
	elif x_45_scale <= 6e-160:
		tmp = -4.0 * (t_1 * (t_0 / y_45_scale))
	elif x_45_scale <= 1.05e+122:
		tmp = t_3
	else:
		tmp = -4.0 * (t_1 * ((b / y_45_scale) * (b / y_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * Float64(b / y_45_scale))
	t_1 = Float64(Float64(a / x_45_scale) * Float64(a / x_45_scale))
	t_2 = Float64(a / Float64(y_45_scale / b))
	t_3 = Float64(-4.0 * Float64(Float64(t_2 * t_2) / Float64(x_45_scale * x_45_scale)))
	tmp = 0.0
	if (x_45_scale <= -1.8e+181)
		tmp = Float64(-4.0 * Float64(Float64((Float64(a / x_45_scale) ^ 2.0) * t_0) / y_45_scale));
	elseif (x_45_scale <= -1.14e-160)
		tmp = t_3;
	elseif (x_45_scale <= 6e-160)
		tmp = Float64(-4.0 * Float64(t_1 * Float64(t_0 / y_45_scale)));
	elseif (x_45_scale <= 1.05e+122)
		tmp = t_3;
	else
		tmp = Float64(-4.0 * Float64(t_1 * Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b * (b / y_45_scale);
	t_1 = (a / x_45_scale) * (a / x_45_scale);
	t_2 = a / (y_45_scale / b);
	t_3 = -4.0 * ((t_2 * t_2) / (x_45_scale * x_45_scale));
	tmp = 0.0;
	if (x_45_scale <= -1.8e+181)
		tmp = -4.0 * ((((a / x_45_scale) ^ 2.0) * t_0) / y_45_scale);
	elseif (x_45_scale <= -1.14e-160)
		tmp = t_3;
	elseif (x_45_scale <= 6e-160)
		tmp = -4.0 * (t_1 * (t_0 / y_45_scale));
	elseif (x_45_scale <= 1.05e+122)
		tmp = t_3;
	else
		tmp = -4.0 * (t_1 * ((b / y_45_scale) * (b / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / x$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.8e+181], N[(-4.0 * N[(N[(N[Power[N[(a / x$45$scale), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, -1.14e-160], t$95$3, If[LessEqual[x$45$scale, 6e-160], N[(-4.0 * N[(t$95$1 * N[(t$95$0 / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 1.05e+122], t$95$3, N[(-4.0 * N[(t$95$1 * N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x-scale < -1.79999999999999992e181

   1. Initial program 34.8%

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

   2. Taylor expanded in angle around 0 34.6%

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

      1. *-commutative 34.6%

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

      2. times-frac 30.9%

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

      3. unpow2 30.9%

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

      4. unpow2 30.9%

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

      5. unpow2 30.9%

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

      6. unpow2 30.9%

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

   4. Simplified 30.9%

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

      1. pow1 30.9%

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

      2. times-frac 55.3%

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

      3. times-frac 62.8%

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

   6. Applied egg-rr 62.8%

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

      1. associate-*r/ 62.8%

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

   8. Applied egg-rr 62.8%

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

      1. associate-*r/ 76.7%

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

      2. pow2 76.7%

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

      3. *-commutative 76.7%

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

   10. Applied egg-rr 76.7%

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

   if -1.79999999999999992e181 < x-scale < -1.14000000000000004e-160 or 5.99999999999999993e-160 < x-scale < 1.05000000000000008e122

   1. Initial program 23.0%

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

   2. Taylor expanded in angle around 0 54.1%

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

      1. *-commutative 54.1%

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

      2. times-frac 53.4%

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

      3. unpow2 53.4%

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

      4. unpow2 53.4%

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

      5. unpow2 53.4%

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

      6. unpow2 53.4%

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

   4. Simplified 53.4%

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

      1. associate-*l/ 54.7%

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

      2. times-frac 72.4%

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

   6. Applied egg-rr 72.4%

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

   7. Taylor expanded in a around 0 55.5%

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

      1. unpow2 55.5%

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

      2. unpow2 55.5%

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

      3. unpow2 55.5%

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

      4. associate-/l* 54.7%

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

      5. times-frac 72.4%

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

   9. Simplified 72.4%

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

       1. times-frac 90.2%

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

   11. Applied egg-rr 90.2%

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

   if -1.14000000000000004e-160 < x-scale < 5.99999999999999993e-160

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0 40.5%

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

      1. *-commutative 40.5%

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

      2. times-frac 40.6%

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

      3. unpow2 40.6%

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

      4. unpow2 40.6%

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

      5. unpow2 40.6%

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

      6. unpow2 40.6%

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

   4. Simplified 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. times-frac 62.8%

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

   6. Applied egg-rr 62.8%

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

      1. *-lft-identity 62.8%

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

   8. Simplified 62.8%

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

      1. *-un-lft-identity 62.8%

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

      2. times-frac 83.3%

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

   10. Applied egg-rr 83.3%

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

       1. *-lft-identity 83.3%

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

   12. Simplified 83.3%

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

       1. associate-*r/ 83.3%

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

   14. Applied egg-rr 83.3%

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

   if 1.05000000000000008e122 < x-scale

   1. Initial program 34.2%

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

   2. Taylor expanded in angle around 0 44.6%

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

      1. *-commutative 44.6%

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

      2. times-frac 44.6%

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

      3. unpow2 44.6%

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

      4. unpow2 44.6%

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

      5. unpow2 44.6%

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

      6. unpow2 44.6%

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

   4. Simplified 44.6%

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

      1. *-un-lft-identity 44.6%

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

      2. times-frac 61.5%

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

   6. Applied egg-rr 61.5%

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

      1. *-lft-identity 61.5%

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

   8. Simplified 61.5%

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

      1. *-un-lft-identity 61.5%

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

      2. times-frac 86.2%

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

   10. Applied egg-rr 86.2%

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

       1. *-lft-identity 86.2%

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

   12. Simplified 86.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.8 \cdot 10^{+181}:\\ \;\;\;\;-4 \cdot \frac{{\left(\frac{a}{x-scale}\right)}^{2} \cdot \left(b \cdot \frac{b}{y-scale}\right)}{y-scale}\\ \mathbf{elif}\;x-scale \leq -1.14 \cdot 10^{-160}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{\frac{y-scale}{b}} \cdot \frac{a}{\frac{y-scale}{b}}}{x-scale \cdot x-scale}\\ \mathbf{elif}\;x-scale \leq 6 \cdot 10^{-160}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b \cdot \frac{b}{y-scale}}{y-scale}\right)\\ \mathbf{elif}\;x-scale \leq 1.05 \cdot 10^{+122}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{\frac{y-scale}{b}} \cdot \frac{a}{\frac{y-scale}{b}}}{x-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 2: 77.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{\frac{y-scale}{b}}\\ \mathbf{if}\;a \leq 4.4 \cdot 10^{-117}:\\ \;\;\;\;-4 \cdot \frac{t_0 \cdot t_0}{x-scale \cdot x-scale}\\ \mathbf{elif}\;a \leq 10^{+116}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot a}{x-scale} \cdot \frac{{\left(\frac{b}{y-scale}\right)}^{2}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (/ y-scale b))))
   (if (<= a 4.4e-117)
     (* -4.0 (/ (* t_0 t_0) (* x-scale x-scale)))
     (if (<= a 1e+116)
       (* -4.0 (* (/ (* a a) x-scale) (/ (pow (/ b y-scale) 2.0) x-scale)))
       (*
        -4.0
        (*
         (* (/ a x-scale) (/ a x-scale))
         (* (/ b y-scale) (/ b y-scale))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale / b);
	double tmp;
	if (a <= 4.4e-117) {
		tmp = -4.0 * ((t_0 * t_0) / (x_45_scale * x_45_scale));
	} else if (a <= 1e+116) {
		tmp = -4.0 * (((a * a) / x_45_scale) * (pow((b / y_45_scale), 2.0) / x_45_scale));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (y_45scale / b)
    if (a <= 4.4d-117) then
        tmp = (-4.0d0) * ((t_0 * t_0) / (x_45scale * x_45scale))
    else if (a <= 1d+116) then
        tmp = (-4.0d0) * (((a * a) / x_45scale) * (((b / y_45scale) ** 2.0d0) / x_45scale))
    else
        tmp = (-4.0d0) * (((a / x_45scale) * (a / x_45scale)) * ((b / y_45scale) * (b / y_45scale)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale / b);
	double tmp;
	if (a <= 4.4e-117) {
		tmp = -4.0 * ((t_0 * t_0) / (x_45_scale * x_45_scale));
	} else if (a <= 1e+116) {
		tmp = -4.0 * (((a * a) / x_45_scale) * (Math.pow((b / y_45_scale), 2.0) / x_45_scale));
	} else {
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a / (y_45_scale / b)
	tmp = 0
	if a <= 4.4e-117:
		tmp = -4.0 * ((t_0 * t_0) / (x_45_scale * x_45_scale))
	elif a <= 1e+116:
		tmp = -4.0 * (((a * a) / x_45_scale) * (math.pow((b / y_45_scale), 2.0) / x_45_scale))
	else:
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale / b))
	tmp = 0.0
	if (a <= 4.4e-117)
		tmp = Float64(-4.0 * Float64(Float64(t_0 * t_0) / Float64(x_45_scale * x_45_scale)));
	elseif (a <= 1e+116)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a * a) / x_45_scale) * Float64((Float64(b / y_45_scale) ^ 2.0) / x_45_scale)));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(a / x_45_scale) * Float64(a / x_45_scale)) * Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a / (y_45_scale / b);
	tmp = 0.0;
	if (a <= 4.4e-117)
		tmp = -4.0 * ((t_0 * t_0) / (x_45_scale * x_45_scale));
	elseif (a <= 1e+116)
		tmp = -4.0 * (((a * a) / x_45_scale) * (((b / y_45_scale) ^ 2.0) / x_45_scale));
	else
		tmp = -4.0 * (((a / x_45_scale) * (a / x_45_scale)) * ((b / y_45_scale) * (b / y_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 4.4e-117], N[(-4.0 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1e+116], N[(-4.0 * N[(N[(N[(a * a), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[Power[N[(b / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(a / x$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.4000000000000002e-117

   1. Initial program 23.8%

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

   2. Taylor expanded in angle around 0 45.4%

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

      1. *-commutative 45.4%

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

      2. times-frac 45.8%

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

      3. unpow2 45.8%

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

      4. unpow2 45.8%

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

      5. unpow2 45.8%

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

      6. unpow2 45.8%

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

   4. Simplified 45.8%

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

      1. associate-*l/ 44.7%

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

      2. times-frac 58.5%

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

   6. Applied egg-rr 58.5%

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

   7. Taylor expanded in a around 0 46.0%

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

      1. unpow2 46.0%

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

      2. unpow2 46.0%

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

      3. unpow2 46.0%

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

      4. associate-/l* 44.6%

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

      5. times-frac 58.4%

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

   9. Simplified 58.4%

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

       1. times-frac 77.3%

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

   11. Applied egg-rr 77.3%

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

   if 4.4000000000000002e-117 < a < 1.00000000000000002e116

   1. Initial program 37.0%

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

   2. Taylor expanded in angle around 0 55.3%

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

      1. *-commutative 55.3%

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

      2. times-frac 50.9%

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

      3. unpow2 50.9%

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

      4. unpow2 50.9%

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

      5. unpow2 50.9%

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

      6. unpow2 50.9%

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

   4. Simplified 50.9%

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

      1. associate-*l/ 55.4%

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

      2. times-frac 71.4%

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

   6. Applied egg-rr 71.4%

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

      1. times-frac 87.9%

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

      2. pow2 87.9%

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

   8. Applied egg-rr 87.9%

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

   if 1.00000000000000002e116 < a

   1. Initial program 4.8%

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

   2. Taylor expanded in angle around 0 45.7%

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

      1. *-commutative 45.7%

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

      2. times-frac 45.7%

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

      3. unpow2 45.7%

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

      4. unpow2 45.7%

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

      5. unpow2 45.7%

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

      6. unpow2 45.7%

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

   4. Simplified 45.7%

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

      1. *-un-lft-identity 45.7%

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

      2. times-frac 60.5%

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

   6. Applied egg-rr 60.5%

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

      1. *-lft-identity 60.5%

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

   8. Simplified 60.5%

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

      1. *-un-lft-identity 60.5%

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

      2. times-frac 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. *-lft-identity 79.4%

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

   12. Simplified 79.4%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.4 \cdot 10^{-117}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{\frac{y-scale}{b}} \cdot \frac{a}{\frac{y-scale}{b}}}{x-scale \cdot x-scale}\\ \mathbf{elif}\;a \leq 10^{+116}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot a}{x-scale} \cdot \frac{{\left(\frac{b}{y-scale}\right)}^{2}}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 3: 82.3% accurate, 98.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale} \cdot \frac{a}{x-scale}\\ t_1 := -4 \cdot \left(t_0 \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ t_2 := \frac{a}{\frac{y-scale}{b}}\\ t_3 := -4 \cdot \frac{t_2 \cdot t_2}{x-scale \cdot x-scale}\\ \mathbf{if}\;x-scale \leq -1.6 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x-scale \leq -3.7 \cdot 10^{-161}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x-scale \leq 5.2 \cdot 10^{-161}:\\ \;\;\;\;-4 \cdot \left(t_0 \cdot \frac{b \cdot \frac{b}{y-scale}}{y-scale}\right)\\ \mathbf{elif}\;x-scale \leq 9 \cdot 10^{+116}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ a x-scale) (/ a x-scale)))
        (t_1 (* -4.0 (* t_0 (* (/ b y-scale) (/ b y-scale)))))
        (t_2 (/ a (/ y-scale b)))
        (t_3 (* -4.0 (/ (* t_2 t_2) (* x-scale x-scale)))))
   (if (<= x-scale -1.6e+206)
     t_1
     (if (<= x-scale -3.7e-161)
       t_3
       (if (<= x-scale 5.2e-161)
         (* -4.0 (* t_0 (/ (* b (/ b y-scale)) y-scale)))
         (if (<= x-scale 9e+116) t_3 t_1))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / x_45_scale) * (a / x_45_scale);
	double t_1 = -4.0 * (t_0 * ((b / y_45_scale) * (b / y_45_scale)));
	double t_2 = a / (y_45_scale / b);
	double t_3 = -4.0 * ((t_2 * t_2) / (x_45_scale * x_45_scale));
	double tmp;
	if (x_45_scale <= -1.6e+206) {
		tmp = t_1;
	} else if (x_45_scale <= -3.7e-161) {
		tmp = t_3;
	} else if (x_45_scale <= 5.2e-161) {
		tmp = -4.0 * (t_0 * ((b * (b / y_45_scale)) / y_45_scale));
	} else if (x_45_scale <= 9e+116) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (a / x_45scale) * (a / x_45scale)
    t_1 = (-4.0d0) * (t_0 * ((b / y_45scale) * (b / y_45scale)))
    t_2 = a / (y_45scale / b)
    t_3 = (-4.0d0) * ((t_2 * t_2) / (x_45scale * x_45scale))
    if (x_45scale <= (-1.6d+206)) then
        tmp = t_1
    else if (x_45scale <= (-3.7d-161)) then
        tmp = t_3
    else if (x_45scale <= 5.2d-161) then
        tmp = (-4.0d0) * (t_0 * ((b * (b / y_45scale)) / y_45scale))
    else if (x_45scale <= 9d+116) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a / x_45_scale) * (a / x_45_scale);
	double t_1 = -4.0 * (t_0 * ((b / y_45_scale) * (b / y_45_scale)));
	double t_2 = a / (y_45_scale / b);
	double t_3 = -4.0 * ((t_2 * t_2) / (x_45_scale * x_45_scale));
	double tmp;
	if (x_45_scale <= -1.6e+206) {
		tmp = t_1;
	} else if (x_45_scale <= -3.7e-161) {
		tmp = t_3;
	} else if (x_45_scale <= 5.2e-161) {
		tmp = -4.0 * (t_0 * ((b * (b / y_45_scale)) / y_45_scale));
	} else if (x_45_scale <= 9e+116) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a / x_45_scale) * (a / x_45_scale)
	t_1 = -4.0 * (t_0 * ((b / y_45_scale) * (b / y_45_scale)))
	t_2 = a / (y_45_scale / b)
	t_3 = -4.0 * ((t_2 * t_2) / (x_45_scale * x_45_scale))
	tmp = 0
	if x_45_scale <= -1.6e+206:
		tmp = t_1
	elif x_45_scale <= -3.7e-161:
		tmp = t_3
	elif x_45_scale <= 5.2e-161:
		tmp = -4.0 * (t_0 * ((b * (b / y_45_scale)) / y_45_scale))
	elif x_45_scale <= 9e+116:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a / x_45_scale) * Float64(a / x_45_scale))
	t_1 = Float64(-4.0 * Float64(t_0 * Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale))))
	t_2 = Float64(a / Float64(y_45_scale / b))
	t_3 = Float64(-4.0 * Float64(Float64(t_2 * t_2) / Float64(x_45_scale * x_45_scale)))
	tmp = 0.0
	if (x_45_scale <= -1.6e+206)
		tmp = t_1;
	elseif (x_45_scale <= -3.7e-161)
		tmp = t_3;
	elseif (x_45_scale <= 5.2e-161)
		tmp = Float64(-4.0 * Float64(t_0 * Float64(Float64(b * Float64(b / y_45_scale)) / y_45_scale)));
	elseif (x_45_scale <= 9e+116)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a / x_45_scale) * (a / x_45_scale);
	t_1 = -4.0 * (t_0 * ((b / y_45_scale) * (b / y_45_scale)));
	t_2 = a / (y_45_scale / b);
	t_3 = -4.0 * ((t_2 * t_2) / (x_45_scale * x_45_scale));
	tmp = 0.0;
	if (x_45_scale <= -1.6e+206)
		tmp = t_1;
	elseif (x_45_scale <= -3.7e-161)
		tmp = t_3;
	elseif (x_45_scale <= 5.2e-161)
		tmp = -4.0 * (t_0 * ((b * (b / y_45_scale)) / y_45_scale));
	elseif (x_45_scale <= 9e+116)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a / x$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(t$95$0 * N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, -1.6e+206], t$95$1, If[LessEqual[x$45$scale, -3.7e-161], t$95$3, If[LessEqual[x$45$scale, 5.2e-161], N[(-4.0 * N[(t$95$0 * N[(N[(b * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 9e+116], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < -1.60000000000000003e206 or 9.00000000000000032e116 < x-scale

   1. Initial program 35.7%

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

   2. Taylor expanded in angle around 0 42.2%

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

      1. *-commutative 42.2%

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

      2. times-frac 40.5%

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

      3. unpow2 40.5%

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

      4. unpow2 40.5%

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

      5. unpow2 40.5%

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

      6. unpow2 40.5%

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

   4. Simplified 40.5%

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

      1. *-un-lft-identity 40.5%

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

      2. times-frac 61.8%

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

   6. Applied egg-rr 61.8%

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

      1. *-lft-identity 61.8%

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

   8. Simplified 61.8%

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

      1. *-un-lft-identity 61.8%

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

      2. times-frac 80.9%

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

   10. Applied egg-rr 80.9%

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

       1. *-lft-identity 80.9%

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

   12. Simplified 80.9%

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

   if -1.60000000000000003e206 < x-scale < -3.6999999999999998e-161 or 5.19999999999999991e-161 < x-scale < 9.00000000000000032e116

   1. Initial program 22.9%

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

   2. Taylor expanded in angle around 0 52.9%

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

      1. *-commutative 52.9%

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

      2. times-frac 52.2%

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

      3. unpow2 52.2%

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

      4. unpow2 52.2%

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

      5. unpow2 52.2%

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

      6. unpow2 52.2%

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

   4. Simplified 52.2%

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

      1. associate-*l/ 53.5%

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

      2. times-frac 70.5%

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

   6. Applied egg-rr 70.5%

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

   7. Taylor expanded in a around 0 54.3%

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

      1. unpow2 54.3%

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

      2. unpow2 54.3%

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

      3. unpow2 54.3%

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

      4. associate-/l* 53.4%

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

      5. times-frac 70.5%

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

   9. Simplified 70.5%

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

       1. times-frac 88.6%

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

   11. Applied egg-rr 88.6%

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

   if -3.6999999999999998e-161 < x-scale < 5.19999999999999991e-161

   1. Initial program 13.7%

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

   2. Taylor expanded in angle around 0 40.5%

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

      1. *-commutative 40.5%

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

      2. times-frac 40.6%

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

      3. unpow2 40.6%

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

      4. unpow2 40.6%

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

      5. unpow2 40.6%

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

      6. unpow2 40.6%

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

   4. Simplified 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. times-frac 62.8%

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

   6. Applied egg-rr 62.8%

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

      1. *-lft-identity 62.8%

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

   8. Simplified 62.8%

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

      1. *-un-lft-identity 62.8%

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

      2. times-frac 83.3%

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

   10. Applied egg-rr 83.3%

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

       1. *-lft-identity 83.3%

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

   12. Simplified 83.3%

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

       1. associate-*r/ 83.3%

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

   14. Applied egg-rr 83.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq -1.6 \cdot 10^{+206}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \mathbf{elif}\;x-scale \leq -3.7 \cdot 10^{-161}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{\frac{y-scale}{b}} \cdot \frac{a}{\frac{y-scale}{b}}}{x-scale \cdot x-scale}\\ \mathbf{elif}\;x-scale \leq 5.2 \cdot 10^{-161}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \frac{b \cdot \frac{b}{y-scale}}{y-scale}\right)\\ \mathbf{elif}\;x-scale \leq 9 \cdot 10^{+116}:\\ \;\;\;\;-4 \cdot \frac{\frac{a}{\frac{y-scale}{b}} \cdot \frac{a}{\frac{y-scale}{b}}}{x-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right) \cdot \left(\frac{b}{y-scale} \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]

Alternative 4: 77.6% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.6%

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

2. Taylor expanded in angle around 0 47.6%

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

   1. *-commutative 47.6%

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

   2. times-frac 46.9%

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

   3. unpow2 46.9%

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

   4. unpow2 46.9%

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

   5. unpow2 46.9%

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

   6. unpow2 46.9%

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

4. Simplified 46.9%

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

   1. *-un-lft-identity 46.9%

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

   2. times-frac 58.9%

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

6. Applied egg-rr 58.9%

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

   1. *-lft-identity 58.9%

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

8. Simplified 58.9%

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

   1. *-un-lft-identity 58.9%

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

   2. times-frac 76.9%

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

10. Applied egg-rr 76.9%

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

    1. *-lft-identity 76.9%

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

12. Simplified 76.9%

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

13. Final simplification 76.9%

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

Alternative 5: 34.8% accurate, 2485.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale) :precision binary64 0.0)

C

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

Fortran

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

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 23.6%

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

   1. fma-neg 24.9%

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

3. Simplified 19.0%

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

4. Taylor expanded in b around 0 22.4%

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

   1. *-commutative 22.4%

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

   2. *-commutative 22.4%

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

   3. *-commutative 22.4%

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

   4. distribute-lft-out 22.4%

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

6. Simplified 34.7%

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

7. Final simplification 34.7%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023257 
(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))))