Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.5% → 78.3%

Time: 1.6min

Alternatives: 7

Speedup: 2485.0×

• 33.6% 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: 24.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.3% accurate, 98.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{b}{y-scale}}{\frac{y-scale}{b}}\\ t_1 := a \cdot \frac{a}{x-scale}\\ \mathbf{if}\;y-scale \leq -4.2 \cdot 10^{+160}:\\ \;\;\;\;-4 \cdot \left(t_1 \cdot \frac{t_0}{x-scale}\right)\\ \mathbf{elif}\;y-scale \leq -8.2 \cdot 10^{-70}:\\ \;\;\;\;-4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot y-scale}}{x-scale \cdot x-scale}\\ \mathbf{elif}\;y-scale \leq 4.9 \cdot 10^{-116}:\\ \;\;\;\;-4 \cdot \frac{t_1 \cdot t_0}{x-scale}\\ \mathbf{elif}\;y-scale \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-4 \cdot \left(\frac{a}{x-scale} \cdot \left(b \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\right)}{y-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}{x-scale} \cdot \frac{a}{\frac{x-scale}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (/ b y-scale) (/ y-scale b))) (t_1 (* a (/ a x-scale))))
   (if (<= y-scale -4.2e+160)
     (* -4.0 (* t_1 (/ t_0 x-scale)))
     (if (<= y-scale -8.2e-70)
       (*
        -4.0
        (/ (/ (* (* b a) (* b a)) (* y-scale y-scale)) (* x-scale x-scale)))
       (if (<= y-scale 4.9e-116)
         (* -4.0 (/ (* t_1 t_0) x-scale))
         (if (<= y-scale 1.35e+154)
           (/
            (* -4.0 (* (/ a x-scale) (* b (* b (/ a x-scale)))))
            (* y-scale y-scale))
           (*
            -4.0
            (*
             (/ (* (/ b y-scale) (/ b y-scale)) x-scale)
             (/ a (/ x-scale a))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b / y_45_scale) / (y_45_scale / b);
	double t_1 = a * (a / x_45_scale);
	double tmp;
	if (y_45_scale <= -4.2e+160) {
		tmp = -4.0 * (t_1 * (t_0 / x_45_scale));
	} else if (y_45_scale <= -8.2e-70) {
		tmp = -4.0 * ((((b * a) * (b * a)) / (y_45_scale * y_45_scale)) / (x_45_scale * x_45_scale));
	} else if (y_45_scale <= 4.9e-116) {
		tmp = -4.0 * ((t_1 * t_0) / x_45_scale);
	} else if (y_45_scale <= 1.35e+154) {
		tmp = (-4.0 * ((a / x_45_scale) * (b * (b * (a / x_45_scale))))) / (y_45_scale * y_45_scale);
	} else {
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)));
	}
	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) :: tmp
    t_0 = (b / y_45scale) / (y_45scale / b)
    t_1 = a * (a / x_45scale)
    if (y_45scale <= (-4.2d+160)) then
        tmp = (-4.0d0) * (t_1 * (t_0 / x_45scale))
    else if (y_45scale <= (-8.2d-70)) then
        tmp = (-4.0d0) * ((((b * a) * (b * a)) / (y_45scale * y_45scale)) / (x_45scale * x_45scale))
    else if (y_45scale <= 4.9d-116) then
        tmp = (-4.0d0) * ((t_1 * t_0) / x_45scale)
    else if (y_45scale <= 1.35d+154) then
        tmp = ((-4.0d0) * ((a / x_45scale) * (b * (b * (a / x_45scale))))) / (y_45scale * y_45scale)
    else
        tmp = (-4.0d0) * ((((b / y_45scale) * (b / y_45scale)) / x_45scale) * (a / (x_45scale / a)))
    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 / y_45_scale) / (y_45_scale / b);
	double t_1 = a * (a / x_45_scale);
	double tmp;
	if (y_45_scale <= -4.2e+160) {
		tmp = -4.0 * (t_1 * (t_0 / x_45_scale));
	} else if (y_45_scale <= -8.2e-70) {
		tmp = -4.0 * ((((b * a) * (b * a)) / (y_45_scale * y_45_scale)) / (x_45_scale * x_45_scale));
	} else if (y_45_scale <= 4.9e-116) {
		tmp = -4.0 * ((t_1 * t_0) / x_45_scale);
	} else if (y_45_scale <= 1.35e+154) {
		tmp = (-4.0 * ((a / x_45_scale) * (b * (b * (a / x_45_scale))))) / (y_45_scale * y_45_scale);
	} else {
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b / y_45_scale) / (y_45_scale / b)
	t_1 = a * (a / x_45_scale)
	tmp = 0
	if y_45_scale <= -4.2e+160:
		tmp = -4.0 * (t_1 * (t_0 / x_45_scale))
	elif y_45_scale <= -8.2e-70:
		tmp = -4.0 * ((((b * a) * (b * a)) / (y_45_scale * y_45_scale)) / (x_45_scale * x_45_scale))
	elif y_45_scale <= 4.9e-116:
		tmp = -4.0 * ((t_1 * t_0) / x_45_scale)
	elif y_45_scale <= 1.35e+154:
		tmp = (-4.0 * ((a / x_45_scale) * (b * (b * (a / x_45_scale))))) / (y_45_scale * y_45_scale)
	else:
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b / y_45_scale) / Float64(y_45_scale / b))
	t_1 = Float64(a * Float64(a / x_45_scale))
	tmp = 0.0
	if (y_45_scale <= -4.2e+160)
		tmp = Float64(-4.0 * Float64(t_1 * Float64(t_0 / x_45_scale)));
	elseif (y_45_scale <= -8.2e-70)
		tmp = Float64(-4.0 * Float64(Float64(Float64(Float64(b * a) * Float64(b * a)) / Float64(y_45_scale * y_45_scale)) / Float64(x_45_scale * x_45_scale)));
	elseif (y_45_scale <= 4.9e-116)
		tmp = Float64(-4.0 * Float64(Float64(t_1 * t_0) / x_45_scale));
	elseif (y_45_scale <= 1.35e+154)
		tmp = Float64(Float64(-4.0 * Float64(Float64(a / x_45_scale) * Float64(b * Float64(b * Float64(a / x_45_scale))))) / Float64(y_45_scale * y_45_scale));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale)) / x_45_scale) * Float64(a / Float64(x_45_scale / a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b / y_45_scale) / (y_45_scale / b);
	t_1 = a * (a / x_45_scale);
	tmp = 0.0;
	if (y_45_scale <= -4.2e+160)
		tmp = -4.0 * (t_1 * (t_0 / x_45_scale));
	elseif (y_45_scale <= -8.2e-70)
		tmp = -4.0 * ((((b * a) * (b * a)) / (y_45_scale * y_45_scale)) / (x_45_scale * x_45_scale));
	elseif (y_45_scale <= 4.9e-116)
		tmp = -4.0 * ((t_1 * t_0) / x_45_scale);
	elseif (y_45_scale <= 1.35e+154)
		tmp = (-4.0 * ((a / x_45_scale) * (b * (b * (a / x_45_scale))))) / (y_45_scale * y_45_scale);
	else
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b / y$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -4.2e+160], N[(-4.0 * N[(t$95$1 * N[(t$95$0 / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, -8.2e-70], N[(-4.0 * N[(N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 4.9e-116], N[(-4.0 * N[(N[(t$95$1 * t$95$0), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 1.35e+154], N[(N[(-4.0 * N[(N[(a / x$45$scale), $MachinePrecision] * N[(b * N[(b * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(a / N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y-scale < -4.19999999999999993e160

   1. Initial program 53.8%

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

   2. Taylor expanded in angle around 0 42.9%

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

      1. *-commutative 42.9%

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

      2. times-frac 43.0%

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

      3. unpow2 43.0%

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

      4. unpow2 43.0%

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

      5. unpow2 43.0%

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

      6. unpow2 43.0%

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

   4. Simplified 43.0%

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

      1. associate-*l/ 43.0%

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

      2. times-frac 78.8%

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

   6. Applied egg-rr 78.8%

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

      1. unpow2 78.8%

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

      2. times-frac 89.2%

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

      3. unpow2 89.2%

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

      4. associate-*r/ 96.3%

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

   8. Simplified 96.3%

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

      1. associate-*r/ 85.9%

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

   10. Applied egg-rr 85.9%

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

       1. *-un-lft-identity 85.9%

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

       2. associate-/l* 96.4%

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

   12. Applied egg-rr 96.4%

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

   if -4.19999999999999993e160 < y-scale < -8.19999999999999955e-70

   1. Initial program 28.1%

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

   2. Taylor expanded in angle around 0 64.6%

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

      1. associate-*r/ 64.6%

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

      2. times-frac 62.3%

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

      3. unpow2 62.3%

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

      4. unpow2 62.3%

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

      5. unpow2 62.3%

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

      6. unpow2 62.3%

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

   4. Simplified 62.3%

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

      1. associate-*l/ 62.3%

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

      2. associate-*l* 71.2%

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

   6. Applied egg-rr 71.2%

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

   7. Taylor expanded in a around 0 62.3%

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

      1. unpow2 62.3%

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

      2. associate-*r* 71.2%

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

      3. unpow2 71.2%

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

      4. unpow2 71.2%

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

      5. times-frac 71.1%

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

      6. unpow2 71.1%

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

      7. *-commutative 71.1%

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

      8. associate-/l* 72.8%

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

      9. unpow2 72.8%

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

   9. Simplified 72.8%

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

   10. Taylor expanded in a around 0 64.6%

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

       1. *-commutative 64.6%

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

       2. associate-/r* 65.6%

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

       3. *-commutative 65.6%

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

       4. unpow2 65.6%

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

       5. unpow2 65.6%

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

       6. unswap-sqr 89.2%

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

       7. unpow2 89.2%

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

       8. unpow2 89.2%

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

   12. Simplified 89.2%

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

   if -8.19999999999999955e-70 < y-scale < 4.89999999999999977e-116

   1. Initial program 17.8%

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

   2. Taylor expanded in angle around 0 39.3%

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

      1. *-commutative 39.3%

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

      2. times-frac 42.6%

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

      3. unpow2 42.6%

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

      4. unpow2 42.6%

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

      5. unpow2 42.6%

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

      6. unpow2 42.6%

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

   4. Simplified 42.6%

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

      1. associate-*l/ 41.6%

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

      2. times-frac 63.6%

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

   6. Applied egg-rr 63.6%

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

      1. unpow2 63.6%

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

      2. times-frac 71.3%

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

      3. unpow2 71.3%

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

      4. associate-*r/ 75.6%

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

   8. Simplified 75.6%

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

      1. associate-*r/ 73.8%

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

   10. Applied egg-rr 73.8%

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

       1. associate-*r/ 75.9%

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

       2. associate-/l* 78.8%

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

   12. Applied egg-rr 78.8%

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

   if 4.89999999999999977e-116 < y-scale < 1.35000000000000003e154

   1. Initial program 19.7%

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

   2. Taylor expanded in angle around 0 56.0%

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

      1. associate-*r/ 56.0%

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

      2. times-frac 56.1%

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

      3. unpow2 56.1%

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

      4. unpow2 56.1%

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

      5. unpow2 56.1%

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

      6. unpow2 56.1%

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

   4. Simplified 56.1%

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

      1. associate-*l/ 56.1%

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

      2. associate-*l* 60.1%

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

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in a around 0 56.1%

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

      1. unpow2 56.1%

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

      2. associate-*r* 60.1%

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

      3. unpow2 60.1%

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

      4. unpow2 60.1%

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

      5. times-frac 65.8%

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

      6. unpow2 65.8%

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

      7. *-commutative 65.8%

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

      8. associate-/l* 67.5%

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

      9. unpow2 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in b around 0 65.8%

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

       1. associate-*l/ 67.5%

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

       2. unpow2 67.5%

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

       3. associate-*r* 83.3%

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

   12. Simplified 83.3%

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

   if 1.35000000000000003e154 < y-scale

   1. Initial program 60.2%

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

   2. Taylor expanded in angle around 0 48.3%

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

      1. *-commutative 48.3%

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

      2. times-frac 52.1%

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

      3. unpow2 52.1%

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

      4. unpow2 52.1%

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

      5. unpow2 52.1%

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

      6. unpow2 52.1%

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

   4. Simplified 52.1%

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

      1. associate-*l/ 52.3%

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

      2. times-frac 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. unpow2 64.6%

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

      2. times-frac 76.5%

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

      3. unpow2 76.5%

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

      4. associate-*r/ 85.4%

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

   8. Simplified 85.4%

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

   9. Taylor expanded in a around 0 76.5%

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

       1. unpow2 76.5%

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

       2. associate-/l* 85.4%

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

   11. Simplified 85.4%

       \[\leadsto -4 \cdot \left(\color{blue}{\frac{a}{\frac{x-scale}{a}}} \cdot \frac{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}{x-scale}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -4.2 \cdot 10^{+160}:\\ \;\;\;\;-4 \cdot \left(\left(a \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{\frac{b}{y-scale}}{\frac{y-scale}{b}}}{x-scale}\right)\\ \mathbf{elif}\;y-scale \leq -8.2 \cdot 10^{-70}:\\ \;\;\;\;-4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot y-scale}}{x-scale \cdot x-scale}\\ \mathbf{elif}\;y-scale \leq 4.9 \cdot 10^{-116}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{b}{y-scale}}{\frac{y-scale}{b}}}{x-scale}\\ \mathbf{elif}\;y-scale \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-4 \cdot \left(\frac{a}{x-scale} \cdot \left(b \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\right)}{y-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}{x-scale} \cdot \frac{a}{\frac{x-scale}{a}}\right)\\ \end{array} \]

Alternative 2: 78.4% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-112}:\\ \;\;\;\;-4 \cdot \left({\left(\frac{b}{y-scale}\right)}^{2} \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-13}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a \cdot \left(\frac{a}{x-scale} \cdot \frac{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}{x-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 6.2e-112)
   (* -4.0 (* (pow (/ b y-scale) 2.0) (* (/ a x-scale) (/ a x-scale))))
   (if (<= b 4.1e-13)
     (*
      -4.0
      (/ (* a (* a (* b b))) (* x-scale (* y-scale (* y-scale x-scale)))))
     (*
      -4.0
      (* a (* (/ a x-scale) (/ (* (/ b y-scale) (/ b y-scale)) x-scale)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 6.2e-112) {
		tmp = -4.0 * (pow((b / y_45_scale), 2.0) * ((a / x_45_scale) * (a / x_45_scale)));
	} else if (b <= 4.1e-13) {
		tmp = -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale))));
	} else {
		tmp = -4.0 * (a * ((a / x_45_scale) * (((b / y_45_scale) * (b / y_45_scale)) / x_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) :: tmp
    if (b <= 6.2d-112) then
        tmp = (-4.0d0) * (((b / y_45scale) ** 2.0d0) * ((a / x_45scale) * (a / x_45scale)))
    else if (b <= 4.1d-13) then
        tmp = (-4.0d0) * ((a * (a * (b * b))) / (x_45scale * (y_45scale * (y_45scale * x_45scale))))
    else
        tmp = (-4.0d0) * (a * ((a / x_45scale) * (((b / y_45scale) * (b / y_45scale)) / x_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 tmp;
	if (b <= 6.2e-112) {
		tmp = -4.0 * (Math.pow((b / y_45_scale), 2.0) * ((a / x_45_scale) * (a / x_45_scale)));
	} else if (b <= 4.1e-13) {
		tmp = -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale))));
	} else {
		tmp = -4.0 * (a * ((a / x_45_scale) * (((b / y_45_scale) * (b / y_45_scale)) / x_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 6.2e-112:
		tmp = -4.0 * (math.pow((b / y_45_scale), 2.0) * ((a / x_45_scale) * (a / x_45_scale)))
	elif b <= 4.1e-13:
		tmp = -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale))))
	else:
		tmp = -4.0 * (a * ((a / x_45_scale) * (((b / y_45_scale) * (b / y_45_scale)) / x_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 6.2e-112)
		tmp = Float64(-4.0 * Float64((Float64(b / y_45_scale) ^ 2.0) * Float64(Float64(a / x_45_scale) * Float64(a / x_45_scale))));
	elseif (b <= 4.1e-13)
		tmp = Float64(-4.0 * Float64(Float64(a * Float64(a * Float64(b * b))) / Float64(x_45_scale * Float64(y_45_scale * Float64(y_45_scale * x_45_scale)))));
	else
		tmp = Float64(-4.0 * Float64(a * Float64(Float64(a / x_45_scale) * Float64(Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale)) / x_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 6.2e-112)
		tmp = -4.0 * (((b / y_45_scale) ^ 2.0) * ((a / x_45_scale) * (a / x_45_scale)));
	elseif (b <= 4.1e-13)
		tmp = -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale))));
	else
		tmp = -4.0 * (a * ((a / x_45_scale) * (((b / y_45_scale) * (b / y_45_scale)) / x_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 6.2e-112], N[(-4.0 * N[(N[Power[N[(b / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(a / x$45$scale), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.1e-13], N[(-4.0 * N[(N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(y$45$scale * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(a * N[(N[(a / x$45$scale), $MachinePrecision] * N[(N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.1999999999999995e-112

   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 45.8%

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

      1. *-commutative 45.8%

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

      2. times-frac 47.8%

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

      3. unpow2 47.8%

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

      4. unpow2 47.8%

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

      5. unpow2 47.8%

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

      6. unpow2 47.8%

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

   4. Simplified 47.8%

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

      1. associate-*l/ 47.9%

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

      2. times-frac 63.7%

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

   6. Applied egg-rr 63.7%

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

      1. unpow2 63.7%

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

      2. times-frac 69.6%

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

      3. unpow2 69.6%

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

      4. associate-*r/ 76.3%

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

   8. Simplified 76.3%

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

   9. Taylor expanded in a around 0 45.8%

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

       1. *-commutative 45.8%

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

       2. times-frac 47.8%

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

       3. unpow2 47.8%

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

       4. unpow2 47.8%

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

       5. times-frac 63.7%

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

       6. unpow2 63.7%

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

       7. unpow2 63.7%

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

       8. unpow2 63.7%

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

       9. times-frac 79.1%

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

   11. Simplified 79.1%

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

   if 6.1999999999999995e-112 < b < 4.1000000000000002e-13

   1. Initial program 31.4%

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

   2. Taylor expanded in angle around 0 75.1%

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

      1. associate-*r/ 75.1%

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

      2. times-frac 74.6%

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

      3. unpow2 74.6%

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

      4. unpow2 74.6%

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

      5. unpow2 74.6%

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

      6. unpow2 74.6%

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

   4. Simplified 74.6%

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

   5. Taylor expanded in y-scale around 0 75.1%

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

      1. unpow2 75.1%

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

      2. associate-*r* 78.7%

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

      3. unpow2 78.7%

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

      4. unpow2 78.7%

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

      5. unpow2 78.7%

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

      6. unpow2 78.7%

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

      7. associate-*l* 85.0%

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

      8. unpow2 85.0%

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

   7. Simplified 85.0%

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

   8. Taylor expanded in x-scale around 0 85.0%

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

      1. *-commutative 85.0%

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

      2. unpow2 85.0%

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

      3. associate-*r* 89.5%

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

      4. *-commutative 89.5%

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

      5. *-commutative 89.5%

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

   10. Simplified 89.5%

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

   if 4.1000000000000002e-13 < b

   1. Initial program 9.5%

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

   2. Taylor expanded in angle around 0 51.0%

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

      1. *-commutative 51.0%

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

      2. times-frac 49.3%

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

      3. unpow2 49.3%

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

      4. unpow2 49.3%

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

      5. unpow2 49.3%

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

      6. unpow2 49.3%

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

   4. Simplified 49.3%

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

      1. associate-*l/ 51.0%

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

      2. times-frac 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. unpow2 69.2%

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

      2. times-frac 73.1%

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

      3. unpow2 73.1%

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

      4. associate-*r/ 82.4%

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

   8. Simplified 82.4%

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

      1. pow1 82.4%

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

      2. associate-*l* 86.3%

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

      3. pow2 86.3%

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

   10. Applied egg-rr 86.3%

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

       1. unpow2 86.3%

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

   12. Applied egg-rr 86.3%

       \[\leadsto -4 \cdot {\left(a \cdot \left(\frac{a}{x-scale} \cdot \frac{\color{blue}{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}}{x-scale}\right)\right)}^{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-112}:\\ \;\;\;\;-4 \cdot \left({\left(\frac{b}{y-scale}\right)}^{2} \cdot \left(\frac{a}{x-scale} \cdot \frac{a}{x-scale}\right)\right)\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-13}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(a \cdot \left(\frac{a}{x-scale} \cdot \frac{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}{x-scale}\right)\right)\\ \end{array} \]

Alternative 3: 78.3% accurate, 98.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{\left(a \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{b}{y-scale}}{\frac{y-scale}{b}}}{x-scale}\\ \mathbf{if}\;y-scale \leq -4.2 \cdot 10^{+160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq -2.2 \cdot 10^{-69}:\\ \;\;\;\;-4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot y-scale}}{x-scale \cdot x-scale}\\ \mathbf{elif}\;y-scale \leq 4.4 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y-scale \leq 1.22 \cdot 10^{+154}:\\ \;\;\;\;\frac{-4 \cdot \left(\frac{a}{x-scale} \cdot \left(b \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\right)}{y-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}{x-scale} \cdot \frac{a}{\frac{x-scale}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          -4.0
          (/
           (* (* a (/ a x-scale)) (/ (/ b y-scale) (/ y-scale b)))
           x-scale))))
   (if (<= y-scale -4.2e+160)
     t_0
     (if (<= y-scale -2.2e-69)
       (*
        -4.0
        (/ (/ (* (* b a) (* b a)) (* y-scale y-scale)) (* x-scale x-scale)))
       (if (<= y-scale 4.4e-116)
         t_0
         (if (<= y-scale 1.22e+154)
           (/
            (* -4.0 (* (/ a x-scale) (* b (* b (/ a x-scale)))))
            (* y-scale y-scale))
           (*
            -4.0
            (*
             (/ (* (/ b y-scale) (/ b y-scale)) x-scale)
             (/ a (/ x-scale a))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = -4.0 * (((a * (a / x_45_scale)) * ((b / y_45_scale) / (y_45_scale / b))) / x_45_scale);
	double tmp;
	if (y_45_scale <= -4.2e+160) {
		tmp = t_0;
	} else if (y_45_scale <= -2.2e-69) {
		tmp = -4.0 * ((((b * a) * (b * a)) / (y_45_scale * y_45_scale)) / (x_45_scale * x_45_scale));
	} else if (y_45_scale <= 4.4e-116) {
		tmp = t_0;
	} else if (y_45_scale <= 1.22e+154) {
		tmp = (-4.0 * ((a / x_45_scale) * (b * (b * (a / x_45_scale))))) / (y_45_scale * y_45_scale);
	} else {
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)));
	}
	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 = (-4.0d0) * (((a * (a / x_45scale)) * ((b / y_45scale) / (y_45scale / b))) / x_45scale)
    if (y_45scale <= (-4.2d+160)) then
        tmp = t_0
    else if (y_45scale <= (-2.2d-69)) then
        tmp = (-4.0d0) * ((((b * a) * (b * a)) / (y_45scale * y_45scale)) / (x_45scale * x_45scale))
    else if (y_45scale <= 4.4d-116) then
        tmp = t_0
    else if (y_45scale <= 1.22d+154) then
        tmp = ((-4.0d0) * ((a / x_45scale) * (b * (b * (a / x_45scale))))) / (y_45scale * y_45scale)
    else
        tmp = (-4.0d0) * ((((b / y_45scale) * (b / y_45scale)) / x_45scale) * (a / (x_45scale / a)))
    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 = -4.0 * (((a * (a / x_45_scale)) * ((b / y_45_scale) / (y_45_scale / b))) / x_45_scale);
	double tmp;
	if (y_45_scale <= -4.2e+160) {
		tmp = t_0;
	} else if (y_45_scale <= -2.2e-69) {
		tmp = -4.0 * ((((b * a) * (b * a)) / (y_45_scale * y_45_scale)) / (x_45_scale * x_45_scale));
	} else if (y_45_scale <= 4.4e-116) {
		tmp = t_0;
	} else if (y_45_scale <= 1.22e+154) {
		tmp = (-4.0 * ((a / x_45_scale) * (b * (b * (a / x_45_scale))))) / (y_45_scale * y_45_scale);
	} else {
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = -4.0 * (((a * (a / x_45_scale)) * ((b / y_45_scale) / (y_45_scale / b))) / x_45_scale)
	tmp = 0
	if y_45_scale <= -4.2e+160:
		tmp = t_0
	elif y_45_scale <= -2.2e-69:
		tmp = -4.0 * ((((b * a) * (b * a)) / (y_45_scale * y_45_scale)) / (x_45_scale * x_45_scale))
	elif y_45_scale <= 4.4e-116:
		tmp = t_0
	elif y_45_scale <= 1.22e+154:
		tmp = (-4.0 * ((a / x_45_scale) * (b * (b * (a / x_45_scale))))) / (y_45_scale * y_45_scale)
	else:
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(-4.0 * Float64(Float64(Float64(a * Float64(a / x_45_scale)) * Float64(Float64(b / y_45_scale) / Float64(y_45_scale / b))) / x_45_scale))
	tmp = 0.0
	if (y_45_scale <= -4.2e+160)
		tmp = t_0;
	elseif (y_45_scale <= -2.2e-69)
		tmp = Float64(-4.0 * Float64(Float64(Float64(Float64(b * a) * Float64(b * a)) / Float64(y_45_scale * y_45_scale)) / Float64(x_45_scale * x_45_scale)));
	elseif (y_45_scale <= 4.4e-116)
		tmp = t_0;
	elseif (y_45_scale <= 1.22e+154)
		tmp = Float64(Float64(-4.0 * Float64(Float64(a / x_45_scale) * Float64(b * Float64(b * Float64(a / x_45_scale))))) / Float64(y_45_scale * y_45_scale));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale)) / x_45_scale) * Float64(a / Float64(x_45_scale / a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = -4.0 * (((a * (a / x_45_scale)) * ((b / y_45_scale) / (y_45_scale / b))) / x_45_scale);
	tmp = 0.0;
	if (y_45_scale <= -4.2e+160)
		tmp = t_0;
	elseif (y_45_scale <= -2.2e-69)
		tmp = -4.0 * ((((b * a) * (b * a)) / (y_45_scale * y_45_scale)) / (x_45_scale * x_45_scale));
	elseif (y_45_scale <= 4.4e-116)
		tmp = t_0;
	elseif (y_45_scale <= 1.22e+154)
		tmp = (-4.0 * ((a / x_45_scale) * (b * (b * (a / x_45_scale))))) / (y_45_scale * y_45_scale);
	else
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(-4.0 * N[(N[(N[(a * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / y$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, -4.2e+160], t$95$0, If[LessEqual[y$45$scale, -2.2e-69], N[(-4.0 * N[(N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale, 4.4e-116], t$95$0, If[LessEqual[y$45$scale, 1.22e+154], N[(N[(-4.0 * N[(N[(a / x$45$scale), $MachinePrecision] * N[(b * N[(b * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(a / N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y-scale < -4.19999999999999993e160 or -2.2e-69 < y-scale < 4.4000000000000002e-116

   1. Initial program 26.3%

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

   2. Taylor expanded in angle around 0 40.2%

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

      1. *-commutative 40.2%

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

      2. times-frac 42.7%

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

      3. unpow2 42.7%

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

      4. unpow2 42.7%

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

      5. unpow2 42.7%

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

      6. unpow2 42.7%

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

   4. Simplified 42.7%

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

      1. associate-*l/ 41.9%

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

      2. times-frac 67.2%

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

   6. Applied egg-rr 67.2%

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

      1. unpow2 67.2%

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

      2. times-frac 75.5%

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

      3. unpow2 75.5%

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

      4. associate-*r/ 80.5%

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

   8. Simplified 80.5%

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

      1. associate-*r/ 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. associate-*r/ 78.3%

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

       2. associate-/l* 83.0%

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

   12. Applied egg-rr 83.0%

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

   if -4.19999999999999993e160 < y-scale < -2.2e-69

   1. Initial program 28.1%

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

   2. Taylor expanded in angle around 0 64.6%

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

      1. associate-*r/ 64.6%

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

      2. times-frac 62.3%

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

      3. unpow2 62.3%

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

      4. unpow2 62.3%

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

      5. unpow2 62.3%

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

      6. unpow2 62.3%

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

   4. Simplified 62.3%

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

      1. associate-*l/ 62.3%

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

      2. associate-*l* 71.2%

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

   6. Applied egg-rr 71.2%

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

   7. Taylor expanded in a around 0 62.3%

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

      1. unpow2 62.3%

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

      2. associate-*r* 71.2%

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

      3. unpow2 71.2%

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

      4. unpow2 71.2%

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

      5. times-frac 71.1%

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

      6. unpow2 71.1%

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

      7. *-commutative 71.1%

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

      8. associate-/l* 72.8%

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

      9. unpow2 72.8%

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

   9. Simplified 72.8%

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

   10. Taylor expanded in a around 0 64.6%

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

       1. *-commutative 64.6%

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

       2. associate-/r* 65.6%

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

       3. *-commutative 65.6%

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

       4. unpow2 65.6%

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

       5. unpow2 65.6%

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

       6. unswap-sqr 89.2%

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

       7. unpow2 89.2%

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

       8. unpow2 89.2%

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

   12. Simplified 89.2%

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

   if 4.4000000000000002e-116 < y-scale < 1.22e154

   1. Initial program 19.7%

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

   2. Taylor expanded in angle around 0 56.0%

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

      1. associate-*r/ 56.0%

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

      2. times-frac 56.1%

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

      3. unpow2 56.1%

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

      4. unpow2 56.1%

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

      5. unpow2 56.1%

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

      6. unpow2 56.1%

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

   4. Simplified 56.1%

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

      1. associate-*l/ 56.1%

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

      2. associate-*l* 60.1%

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

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in a around 0 56.1%

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

      1. unpow2 56.1%

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

      2. associate-*r* 60.1%

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

      3. unpow2 60.1%

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

      4. unpow2 60.1%

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

      5. times-frac 65.8%

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

      6. unpow2 65.8%

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

      7. *-commutative 65.8%

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

      8. associate-/l* 67.5%

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

      9. unpow2 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in b around 0 65.8%

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

       1. associate-*l/ 67.5%

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

       2. unpow2 67.5%

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

       3. associate-*r* 83.3%

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

   12. Simplified 83.3%

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

   if 1.22e154 < y-scale

   1. Initial program 60.2%

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

   2. Taylor expanded in angle around 0 48.3%

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

      1. *-commutative 48.3%

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

      2. times-frac 52.1%

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

      3. unpow2 52.1%

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

      4. unpow2 52.1%

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

      5. unpow2 52.1%

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

      6. unpow2 52.1%

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

   4. Simplified 52.1%

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

      1. associate-*l/ 52.3%

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

      2. times-frac 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. unpow2 64.6%

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

      2. times-frac 76.5%

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

      3. unpow2 76.5%

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

      4. associate-*r/ 85.4%

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

   8. Simplified 85.4%

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

   9. Taylor expanded in a around 0 76.5%

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

       1. unpow2 76.5%

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

       2. associate-/l* 85.4%

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

   11. Simplified 85.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq -4.2 \cdot 10^{+160}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{b}{y-scale}}{\frac{y-scale}{b}}}{x-scale}\\ \mathbf{elif}\;y-scale \leq -2.2 \cdot 10^{-69}:\\ \;\;\;\;-4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot y-scale}}{x-scale \cdot x-scale}\\ \mathbf{elif}\;y-scale \leq 4.4 \cdot 10^{-116}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{b}{y-scale}}{\frac{y-scale}{b}}}{x-scale}\\ \mathbf{elif}\;y-scale \leq 1.22 \cdot 10^{+154}:\\ \;\;\;\;\frac{-4 \cdot \left(\frac{a}{x-scale} \cdot \left(b \cdot \left(b \cdot \frac{a}{x-scale}\right)\right)\right)}{y-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}{x-scale} \cdot \frac{a}{\frac{x-scale}{a}}\right)\\ \end{array} \]

Alternative 4: 76.3% accurate, 117.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{b}{y-scale}}{\frac{y-scale}{b}}}{x-scale}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}{x-scale} \cdot \frac{a}{\frac{x-scale}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 1.22e-119)
   (* -4.0 (/ (* (* a (/ a x-scale)) (/ (/ b y-scale) (/ y-scale b))) x-scale))
   (if (<= b 6.2e+48)
     (*
      -4.0
      (/ (* a (* a (* b b))) (* x-scale (* y-scale (* y-scale x-scale)))))
     (*
      -4.0
      (* (/ (* (/ b y-scale) (/ b y-scale)) x-scale) (/ a (/ x-scale a)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 1.22e-119) {
		tmp = -4.0 * (((a * (a / x_45_scale)) * ((b / y_45_scale) / (y_45_scale / b))) / x_45_scale);
	} else if (b <= 6.2e+48) {
		tmp = -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale))));
	} else {
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)));
	}
	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) :: tmp
    if (b <= 1.22d-119) then
        tmp = (-4.0d0) * (((a * (a / x_45scale)) * ((b / y_45scale) / (y_45scale / b))) / x_45scale)
    else if (b <= 6.2d+48) then
        tmp = (-4.0d0) * ((a * (a * (b * b))) / (x_45scale * (y_45scale * (y_45scale * x_45scale))))
    else
        tmp = (-4.0d0) * ((((b / y_45scale) * (b / y_45scale)) / x_45scale) * (a / (x_45scale / a)))
    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 tmp;
	if (b <= 1.22e-119) {
		tmp = -4.0 * (((a * (a / x_45_scale)) * ((b / y_45_scale) / (y_45_scale / b))) / x_45_scale);
	} else if (b <= 6.2e+48) {
		tmp = -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale))));
	} else {
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 1.22e-119:
		tmp = -4.0 * (((a * (a / x_45_scale)) * ((b / y_45_scale) / (y_45_scale / b))) / x_45_scale)
	elif b <= 6.2e+48:
		tmp = -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale))))
	else:
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 1.22e-119)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a * Float64(a / x_45_scale)) * Float64(Float64(b / y_45_scale) / Float64(y_45_scale / b))) / x_45_scale));
	elseif (b <= 6.2e+48)
		tmp = Float64(-4.0 * Float64(Float64(a * Float64(a * Float64(b * b))) / Float64(x_45_scale * Float64(y_45_scale * Float64(y_45_scale * x_45_scale)))));
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(Float64(b / y_45_scale) * Float64(b / y_45_scale)) / x_45_scale) * Float64(a / Float64(x_45_scale / a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 1.22e-119)
		tmp = -4.0 * (((a * (a / x_45_scale)) * ((b / y_45_scale) / (y_45_scale / b))) / x_45_scale);
	elseif (b <= 6.2e+48)
		tmp = -4.0 * ((a * (a * (b * b))) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale))));
	else
		tmp = -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a / (x_45_scale / a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 1.22e-119], N[(-4.0 * N[(N[(N[(a * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / y$45$scale), $MachinePrecision] / N[(y$45$scale / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e+48], N[(-4.0 * N[(N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(y$45$scale * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(N[(N[(N[(b / y$45$scale), $MachinePrecision] * N[(b / y$45$scale), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(a / N[(x$45$scale / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.22e-119

   1. Initial program 34.4%

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

   2. Taylor expanded in angle around 0 46.1%

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

      1. *-commutative 46.1%

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

      2. times-frac 48.1%

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

      3. unpow2 48.1%

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

      4. unpow2 48.1%

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

      5. unpow2 48.1%

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

      6. unpow2 48.1%

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

   4. Simplified 48.1%

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

      1. associate-*l/ 48.1%

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

      2. times-frac 64.1%

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

   6. Applied egg-rr 64.1%

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

      1. unpow2 64.1%

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

      2. times-frac 70.0%

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

      3. unpow2 70.0%

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

      4. associate-*r/ 76.8%

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

   8. Simplified 76.8%

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

      1. associate-*r/ 73.2%

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

   10. Applied egg-rr 73.2%

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

       1. associate-*r/ 72.4%

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

       2. associate-/l* 76.1%

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

   12. Applied egg-rr 76.1%

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

   if 1.22e-119 < b < 6.20000000000000011e48

   1. Initial program 28.9%

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

   2. Taylor expanded in angle around 0 73.2%

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

      1. associate-*r/ 73.2%

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

      2. times-frac 70.4%

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

      3. unpow2 70.4%

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

      4. unpow2 70.4%

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

      5. unpow2 70.4%

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

      6. unpow2 70.4%

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

   4. Simplified 70.4%

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

   5. Taylor expanded in y-scale around 0 73.2%

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

      1. unpow2 73.2%

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

      2. associate-*r* 75.6%

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

      3. unpow2 75.6%

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

      4. unpow2 75.6%

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

      5. unpow2 75.6%

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

      6. unpow2 75.6%

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

      7. associate-*l* 79.7%

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

      8. unpow2 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in x-scale around 0 79.7%

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

      1. *-commutative 79.7%

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

      2. unpow2 79.7%

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

      3. associate-*r* 85.3%

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

      4. *-commutative 85.3%

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

      5. *-commutative 85.3%

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

   10. Simplified 85.3%

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

   if 6.20000000000000011e48 < b

   1. Initial program 4.9%

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

   2. Taylor expanded in angle around 0 44.2%

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

      1. *-commutative 44.2%

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

      2. times-frac 44.3%

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

      3. unpow2 44.3%

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

      4. unpow2 44.3%

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

      5. unpow2 44.3%

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

      6. unpow2 44.3%

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

   4. Simplified 44.3%

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

      1. associate-*l/ 44.3%

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

      2. times-frac 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. unpow2 67.6%

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

      2. times-frac 74.8%

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

      3. unpow2 74.8%

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

      4. associate-*r/ 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in a around 0 74.8%

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

       1. unpow2 74.8%

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

       2. associate-/l* 85.9%

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

   11. Simplified 85.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;-4 \cdot \frac{\left(a \cdot \frac{a}{x-scale}\right) \cdot \frac{\frac{b}{y-scale}}{\frac{y-scale}{b}}}{x-scale}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{\frac{b}{y-scale} \cdot \frac{b}{y-scale}}{x-scale} \cdot \frac{a}{\frac{x-scale}{a}}\right)\\ \end{array} \]

Alternative 5: 76.1% accurate, 146.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return -4.0 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a * (a / x_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) * ((((b / y_45scale) * (b / y_45scale)) / x_45scale) * (a * (a / x_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 * ((((b / y_45_scale) * (b / y_45_scale)) / x_45_scale) * (a * (a / x_45_scale)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

2. Taylor expanded in angle around 0 49.9%

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

   1. *-commutative 49.9%

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

   2. times-frac 49.3%

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

   3. unpow2 49.3%

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

   4. unpow2 49.3%

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

   5. unpow2 49.3%

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

   6. unpow2 49.3%

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

4. Simplified 49.3%

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

   1. associate-*l/ 49.8%

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

   2. times-frac 64.5%

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

6. Applied egg-rr 64.5%

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

   1. unpow2 64.5%

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

   2. times-frac 70.1%

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

   3. unpow2 70.1%

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

   4. associate-*r/ 77.1%

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

8. Simplified 77.1%

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

9. Final simplification 77.1%

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

Alternative 6: 76.1% accurate, 146.2× speedup

Math

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

FPCore

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

C

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

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) * ((((b / y_45scale) * (b / y_45scale)) / x_45scale) * (a / (x_45scale / a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.6%

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

2. Taylor expanded in angle around 0 49.9%

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

   1. *-commutative 49.9%

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

   2. times-frac 49.3%

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

   3. unpow2 49.3%

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

   4. unpow2 49.3%

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

   5. unpow2 49.3%

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

   6. unpow2 49.3%

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

4. Simplified 49.3%

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

   1. associate-*l/ 49.8%

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

   2. times-frac 64.5%

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

6. Applied egg-rr 64.5%

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

   1. unpow2 64.5%

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

   2. times-frac 70.1%

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

   3. unpow2 70.1%

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

   4. associate-*r/ 77.1%

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

8. Simplified 77.1%

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

9. Taylor expanded in a around 0 70.1%

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

    1. unpow2 70.1%

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

    2. associate-/l* 77.1%

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

11. Simplified 77.1%

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

12. Final simplification 77.1%

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

Alternative 7: 35.0% 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 28.6%

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

   1. fma-neg 30.4%

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

3. Simplified 24.1%

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

4. Taylor expanded in b around 0 23.6%

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

   1. *-commutative 23.6%

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

   2. *-commutative 23.6%

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

   3. *-commutative 23.6%

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

   4. distribute-lft-out 23.6%

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

6. Simplified 37.3%

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

7. Final simplification 37.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023215 
(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))))