a from scale-rotated-ellipse

Percentage Accurate: 3.1% → 56.0%

Time: 40.4s

Alternatives: 10

Speedup: 919.0×

• 30.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) + \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

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
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (+
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

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 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

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(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) + sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
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 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
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[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]}, Block[{t$95$4 = 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]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[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], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 3.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) + \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \end{array} \]

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
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (+
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

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 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

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(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) + sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
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 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
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[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]}, Block[{t$95$4 = 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]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[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], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Alternative 1: 56.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle}}\\ \mathbf{if}\;x-scale\_m \leq 1.95 \cdot 10^{+46}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale\_m \cdot {\left({8}^{0.25}\right)}^{2}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin t\_0, \cos t\_0 \cdot b\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(a \cdot a + {\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle))))
   (if (<= x-scale_m 1.95e+46)
     (*
      (* 0.25 (* y-scale_m (pow (pow 8.0 0.25) 2.0)))
      (* (hypot (* a (sin t_0)) (* (cos t_0) b)) (sqrt 2.0)))
     (*
      (* 0.25 (* x-scale_m (sqrt 8.0)))
      (sqrt
       (*
        2.0
        (+
         (* a a)
         (*
          (pow (sin (* 0.005555555555555556 (* PI angle))) 2.0)
          (* b b)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle);
	double tmp;
	if (x_45_scale_m <= 1.95e+46) {
		tmp = (0.25 * (y_45_scale_m * pow(pow(8.0, 0.25), 2.0))) * (hypot((a * sin(t_0)), (cos(t_0) * b)) * sqrt(2.0));
	} else {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * ((a * a) + (pow(sin((0.005555555555555556 * (((double) M_PI) * angle))), 2.0) * (b * b)))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.PI / (180.0 / angle);
	double tmp;
	if (x_45_scale_m <= 1.95e+46) {
		tmp = (0.25 * (y_45_scale_m * Math.pow(Math.pow(8.0, 0.25), 2.0))) * (Math.hypot((a * Math.sin(t_0)), (Math.cos(t_0) * b)) * Math.sqrt(2.0));
	} else {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.sqrt((2.0 * ((a * a) + (Math.pow(Math.sin((0.005555555555555556 * (Math.PI * angle))), 2.0) * (b * b)))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.pi / (180.0 / angle)
	tmp = 0
	if x_45_scale_m <= 1.95e+46:
		tmp = (0.25 * (y_45_scale_m * math.pow(math.pow(8.0, 0.25), 2.0))) * (math.hypot((a * math.sin(t_0)), (math.cos(t_0) * b)) * math.sqrt(2.0))
	else:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.sqrt((2.0 * ((a * a) + (math.pow(math.sin((0.005555555555555556 * (math.pi * angle))), 2.0) * (b * b)))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(pi / Float64(180.0 / angle))
	tmp = 0.0
	if (x_45_scale_m <= 1.95e+46)
		tmp = Float64(Float64(0.25 * Float64(y_45_scale_m * ((8.0 ^ 0.25) ^ 2.0))) * Float64(hypot(Float64(a * sin(t_0)), Float64(cos(t_0) * b)) * sqrt(2.0)));
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * sqrt(Float64(2.0 * Float64(Float64(a * a) + Float64((sin(Float64(0.005555555555555556 * Float64(pi * angle))) ^ 2.0) * Float64(b * b))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = pi / (180.0 / angle);
	tmp = 0.0;
	if (x_45_scale_m <= 1.95e+46)
		tmp = (0.25 * (y_45_scale_m * ((8.0 ^ 0.25) ^ 2.0))) * (hypot((a * sin(t_0)), (cos(t_0) * b)) * sqrt(2.0));
	else
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * ((a * a) + ((sin((0.005555555555555556 * (pi * angle))) ^ 2.0) * (b * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.95e+46], N[(N[(0.25 * N[(y$45$scale$95$m * N[Power[N[Power[8.0, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[t$95$0], $MachinePrecision] * b), $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(a * a), $MachinePrecision] + N[(N[Power[N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.94999999999999997e46

   1. Initial program 3.3%

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

   3. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 22.7%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 23.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 23.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\color{blue}{{\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot 2}\right)\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{{\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)} \cdot \color{blue}{\sqrt{2}}\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left({\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(\left({\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   9. Applied egg-rr 26.3%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \sqrt{2}\right)} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left({8}^{\frac{1}{2}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \color{blue}{b}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left({8}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {8}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \color{blue}{b}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left({\left({8}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \color{blue}{b}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{pow.f64}\left(\left({8}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), 2\right)\right)\right), \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \color{blue}{b}\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{pow.f64}\left(\left({8}^{\frac{1}{4}}\right), 2\right)\right)\right), \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

       6. pow-lowering-pow.f64 26.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(8, \frac{1}{4}\right), 2\right)\right)\right), \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), b\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   11. Applied egg-rr 26.3%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \color{blue}{{\left({8}^{0.25}\right)}^{2}}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \sqrt{2}\right) \]

   if 1.94999999999999997e46 < x-scale

   1. Initial program 2.3%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{1}, 2\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 60.7%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\color{blue}{1}}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.95 \cdot 10^{+46}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot {\left({8}^{0.25}\right)}^{2}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(a \cdot a + {\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle}}\\ \mathbf{if}\;x-scale\_m \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;\left(\mathsf{hypot}\left(a \cdot \sin t\_0, \cos t\_0 \cdot b\right) \cdot \sqrt{2}\right) \cdot \left(0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(a \cdot a + {\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle))))
   (if (<= x-scale_m 1.15e+46)
     (*
      (* (hypot (* a (sin t_0)) (* (cos t_0) b)) (sqrt 2.0))
      (* 0.25 (* y-scale_m (sqrt 8.0))))
     (*
      (* 0.25 (* x-scale_m (sqrt 8.0)))
      (sqrt
       (*
        2.0
        (+
         (* a a)
         (*
          (pow (sin (* 0.005555555555555556 (* PI angle))) 2.0)
          (* b b)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle);
	double tmp;
	if (x_45_scale_m <= 1.15e+46) {
		tmp = (hypot((a * sin(t_0)), (cos(t_0) * b)) * sqrt(2.0)) * (0.25 * (y_45_scale_m * sqrt(8.0)));
	} else {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * ((a * a) + (pow(sin((0.005555555555555556 * (((double) M_PI) * angle))), 2.0) * (b * b)))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.PI / (180.0 / angle);
	double tmp;
	if (x_45_scale_m <= 1.15e+46) {
		tmp = (Math.hypot((a * Math.sin(t_0)), (Math.cos(t_0) * b)) * Math.sqrt(2.0)) * (0.25 * (y_45_scale_m * Math.sqrt(8.0)));
	} else {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.sqrt((2.0 * ((a * a) + (Math.pow(Math.sin((0.005555555555555556 * (Math.PI * angle))), 2.0) * (b * b)))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.pi / (180.0 / angle)
	tmp = 0
	if x_45_scale_m <= 1.15e+46:
		tmp = (math.hypot((a * math.sin(t_0)), (math.cos(t_0) * b)) * math.sqrt(2.0)) * (0.25 * (y_45_scale_m * math.sqrt(8.0)))
	else:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.sqrt((2.0 * ((a * a) + (math.pow(math.sin((0.005555555555555556 * (math.pi * angle))), 2.0) * (b * b)))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(pi / Float64(180.0 / angle))
	tmp = 0.0
	if (x_45_scale_m <= 1.15e+46)
		tmp = Float64(Float64(hypot(Float64(a * sin(t_0)), Float64(cos(t_0) * b)) * sqrt(2.0)) * Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0))));
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * sqrt(Float64(2.0 * Float64(Float64(a * a) + Float64((sin(Float64(0.005555555555555556 * Float64(pi * angle))) ^ 2.0) * Float64(b * b))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = pi / (180.0 / angle);
	tmp = 0.0;
	if (x_45_scale_m <= 1.15e+46)
		tmp = (hypot((a * sin(t_0)), (cos(t_0) * b)) * sqrt(2.0)) * (0.25 * (y_45_scale_m * sqrt(8.0)));
	else
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * ((a * a) + ((sin((0.005555555555555556 * (pi * angle))) ^ 2.0) * (b * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.15e+46], N[(N[(N[Sqrt[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[t$95$0], $MachinePrecision] * b), $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(a * a), $MachinePrecision] + N[(N[Power[N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.15e46

   1. Initial program 3.3%

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

   3. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 22.7%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 23.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 23.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\color{blue}{{\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot 2}\right)\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{{\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)} \cdot \color{blue}{\sqrt{2}}\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left({\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(\left({\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   9. Applied egg-rr 26.3%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \sqrt{2}\right)} \]

   if 1.15e46 < x-scale

   1. Initial program 2.3%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 60.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{1}, 2\right), \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 60.7%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\color{blue}{1}}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;\left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \sqrt{2}\right) \cdot \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(a \cdot a + {\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle}}\\ \mathbf{if}\;x-scale\_m \leq 1.05 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{hypot}\left(a \cdot \sin t\_0, \cos t\_0 \cdot b\right) \cdot \sqrt{2}\right) \cdot \left(0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(a \cdot a\right) \cdot \left(\left(1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot 0.5\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle))))
   (if (<= x-scale_m 1.05e+64)
     (*
      (* (hypot (* a (sin t_0)) (* (cos t_0) b)) (sqrt 2.0))
      (* 0.25 (* y-scale_m (sqrt 8.0))))
     (*
      x-scale_m
      (*
       0.25
       (sqrt
        (*
         16.0
         (+
          (* (* 3.08641975308642e-5 (* angle angle)) (* (* b b) (* PI PI)))
          (*
           (* a a)
           (* (+ 1.0 (cos (* PI (* angle 0.011111111111111112)))) 0.5))))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle);
	double tmp;
	if (x_45_scale_m <= 1.05e+64) {
		tmp = (hypot((a * sin(t_0)), (cos(t_0) * b)) * sqrt(2.0)) * (0.25 * (y_45_scale_m * sqrt(8.0)));
	} else {
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (((double) M_PI) * ((double) M_PI)))) + ((a * a) * ((1.0 + cos((((double) M_PI) * (angle * 0.011111111111111112)))) * 0.5))))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.PI / (180.0 / angle);
	double tmp;
	if (x_45_scale_m <= 1.05e+64) {
		tmp = (Math.hypot((a * Math.sin(t_0)), (Math.cos(t_0) * b)) * Math.sqrt(2.0)) * (0.25 * (y_45_scale_m * Math.sqrt(8.0)));
	} else {
		tmp = x_45_scale_m * (0.25 * Math.sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (Math.PI * Math.PI))) + ((a * a) * ((1.0 + Math.cos((Math.PI * (angle * 0.011111111111111112)))) * 0.5))))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.pi / (180.0 / angle)
	tmp = 0
	if x_45_scale_m <= 1.05e+64:
		tmp = (math.hypot((a * math.sin(t_0)), (math.cos(t_0) * b)) * math.sqrt(2.0)) * (0.25 * (y_45_scale_m * math.sqrt(8.0)))
	else:
		tmp = x_45_scale_m * (0.25 * math.sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (math.pi * math.pi))) + ((a * a) * ((1.0 + math.cos((math.pi * (angle * 0.011111111111111112)))) * 0.5))))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(pi / Float64(180.0 / angle))
	tmp = 0.0
	if (x_45_scale_m <= 1.05e+64)
		tmp = Float64(Float64(hypot(Float64(a * sin(t_0)), Float64(cos(t_0) * b)) * sqrt(2.0)) * Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0))));
	else
		tmp = Float64(x_45_scale_m * Float64(0.25 * sqrt(Float64(16.0 * Float64(Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(Float64(b * b) * Float64(pi * pi))) + Float64(Float64(a * a) * Float64(Float64(1.0 + cos(Float64(pi * Float64(angle * 0.011111111111111112)))) * 0.5)))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = pi / (180.0 / angle);
	tmp = 0.0;
	if (x_45_scale_m <= 1.05e+64)
		tmp = (hypot((a * sin(t_0)), (cos(t_0) * b)) * sqrt(2.0)) * (0.25 * (y_45_scale_m * sqrt(8.0)));
	else
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (pi * pi))) + ((a * a) * ((1.0 + cos((pi * (angle * 0.011111111111111112)))) * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.05e+64], N[(N[(N[Sqrt[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[t$95$0], $MachinePrecision] * b), $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$45$scale$95$m * N[(0.25 * N[Sqrt[N[(16.0 * N[(N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(1.0 + N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.05e64

   1. Initial program 3.2%

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

   3. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 22.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 23.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 23.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\color{blue}{{\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot 2}\right)\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{{\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)} \cdot \color{blue}{\sqrt{2}}\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left({\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(\left({\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   9. Applied egg-rr 25.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \sqrt{2}\right)} \]

   if 1.05e64 < x-scale

   1. Initial program 2.3%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 60.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   7. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

   9. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \left({angle}^{2}\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot angle\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       12. PI-lowering-PI.f64 59.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

   12. Simplified 59.3%

       \[\leadsto \left(\sqrt{16 \cdot \left(\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right)} + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.05 \cdot 10^{+64}:\\ \;\;\;\;\left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \sqrt{2}\right) \cdot \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(a \cdot a\right) \cdot \left(\left(1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot 0.5\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle}}\\ \mathbf{if}\;x-scale\_m \leq 2.4 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{hypot}\left(a \cdot \sin t\_0, \cos t\_0 \cdot b\right) \cdot \left(\sqrt{8} \cdot \left(0.25 \cdot y-scale\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(a \cdot a\right) \cdot \left(\left(1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot 0.5\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle))))
   (if (<= x-scale_m 2.4e+65)
     (*
      (sqrt 2.0)
      (*
       (hypot (* a (sin t_0)) (* (cos t_0) b))
       (* (sqrt 8.0) (* 0.25 y-scale_m))))
     (*
      x-scale_m
      (*
       0.25
       (sqrt
        (*
         16.0
         (+
          (* (* 3.08641975308642e-5 (* angle angle)) (* (* b b) (* PI PI)))
          (*
           (* a a)
           (* (+ 1.0 (cos (* PI (* angle 0.011111111111111112)))) 0.5))))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle);
	double tmp;
	if (x_45_scale_m <= 2.4e+65) {
		tmp = sqrt(2.0) * (hypot((a * sin(t_0)), (cos(t_0) * b)) * (sqrt(8.0) * (0.25 * y_45_scale_m)));
	} else {
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (((double) M_PI) * ((double) M_PI)))) + ((a * a) * ((1.0 + cos((((double) M_PI) * (angle * 0.011111111111111112)))) * 0.5))))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.PI / (180.0 / angle);
	double tmp;
	if (x_45_scale_m <= 2.4e+65) {
		tmp = Math.sqrt(2.0) * (Math.hypot((a * Math.sin(t_0)), (Math.cos(t_0) * b)) * (Math.sqrt(8.0) * (0.25 * y_45_scale_m)));
	} else {
		tmp = x_45_scale_m * (0.25 * Math.sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (Math.PI * Math.PI))) + ((a * a) * ((1.0 + Math.cos((Math.PI * (angle * 0.011111111111111112)))) * 0.5))))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.pi / (180.0 / angle)
	tmp = 0
	if x_45_scale_m <= 2.4e+65:
		tmp = math.sqrt(2.0) * (math.hypot((a * math.sin(t_0)), (math.cos(t_0) * b)) * (math.sqrt(8.0) * (0.25 * y_45_scale_m)))
	else:
		tmp = x_45_scale_m * (0.25 * math.sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (math.pi * math.pi))) + ((a * a) * ((1.0 + math.cos((math.pi * (angle * 0.011111111111111112)))) * 0.5))))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(pi / Float64(180.0 / angle))
	tmp = 0.0
	if (x_45_scale_m <= 2.4e+65)
		tmp = Float64(sqrt(2.0) * Float64(hypot(Float64(a * sin(t_0)), Float64(cos(t_0) * b)) * Float64(sqrt(8.0) * Float64(0.25 * y_45_scale_m))));
	else
		tmp = Float64(x_45_scale_m * Float64(0.25 * sqrt(Float64(16.0 * Float64(Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(Float64(b * b) * Float64(pi * pi))) + Float64(Float64(a * a) * Float64(Float64(1.0 + cos(Float64(pi * Float64(angle * 0.011111111111111112)))) * 0.5)))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = pi / (180.0 / angle);
	tmp = 0.0;
	if (x_45_scale_m <= 2.4e+65)
		tmp = sqrt(2.0) * (hypot((a * sin(t_0)), (cos(t_0) * b)) * (sqrt(8.0) * (0.25 * y_45_scale_m)));
	else
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (pi * pi))) + ((a * a) * ((1.0 + cos((pi * (angle * 0.011111111111111112)))) * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 2.4e+65], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[t$95$0], $MachinePrecision] * b), $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(0.25 * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$45$scale$95$m * N[(0.25 * N[Sqrt[N[(16.0 * N[(N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(1.0 + N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.4000000000000002e65

   1. Initial program 3.2%

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

   3. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 22.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 23.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 23.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\color{blue}{{\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]

   9. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \left(\sqrt{8} \cdot \left(0.25 \cdot y-scale\right)\right)\right)} \]

   if 2.4000000000000002e65 < x-scale

   1. Initial program 2.3%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 60.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   7. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

   9. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \left({angle}^{2}\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot angle\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       12. PI-lowering-PI.f64 59.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

   12. Simplified 59.3%

       \[\leadsto \left(\sqrt{16 \cdot \left(\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right)} + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.4 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \left(\sqrt{8} \cdot \left(0.25 \cdot y-scale\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(a \cdot a\right) \cdot \left(\left(1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot 0.5\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 9 \cdot 10^{+66}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(a \cdot a\right) \cdot \left(\left(1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot 0.5\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 9e+66)
   (*
    (* 0.25 (* y-scale_m (sqrt 8.0)))
    (* (sqrt 2.0) (hypot (* a (sin (/ PI (/ 180.0 angle)))) b)))
   (*
    x-scale_m
    (*
     0.25
     (sqrt
      (*
       16.0
       (+
        (* (* 3.08641975308642e-5 (* angle angle)) (* (* b b) (* PI PI)))
        (*
         (* a a)
         (* (+ 1.0 (cos (* PI (* angle 0.011111111111111112)))) 0.5)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 9e+66) {
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * hypot((a * sin((((double) M_PI) / (180.0 / angle)))), b));
	} else {
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (((double) M_PI) * ((double) M_PI)))) + ((a * a) * ((1.0 + cos((((double) M_PI) * (angle * 0.011111111111111112)))) * 0.5))))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 9e+66) {
		tmp = (0.25 * (y_45_scale_m * Math.sqrt(8.0))) * (Math.sqrt(2.0) * Math.hypot((a * Math.sin((Math.PI / (180.0 / angle)))), b));
	} else {
		tmp = x_45_scale_m * (0.25 * Math.sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (Math.PI * Math.PI))) + ((a * a) * ((1.0 + Math.cos((Math.PI * (angle * 0.011111111111111112)))) * 0.5))))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 9e+66:
		tmp = (0.25 * (y_45_scale_m * math.sqrt(8.0))) * (math.sqrt(2.0) * math.hypot((a * math.sin((math.pi / (180.0 / angle)))), b))
	else:
		tmp = x_45_scale_m * (0.25 * math.sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (math.pi * math.pi))) + ((a * a) * ((1.0 + math.cos((math.pi * (angle * 0.011111111111111112)))) * 0.5))))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 9e+66)
		tmp = Float64(Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(pi / Float64(180.0 / angle)))), b)));
	else
		tmp = Float64(x_45_scale_m * Float64(0.25 * sqrt(Float64(16.0 * Float64(Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(Float64(b * b) * Float64(pi * pi))) + Float64(Float64(a * a) * Float64(Float64(1.0 + cos(Float64(pi * Float64(angle * 0.011111111111111112)))) * 0.5)))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 9e+66)
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * hypot((a * sin((pi / (180.0 / angle)))), b));
	else
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (pi * pi))) + ((a * a) * ((1.0 + cos((pi * (angle * 0.011111111111111112)))) * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 9e+66], N[(N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$45$scale$95$m * N[(0.25 * N[Sqrt[N[(16.0 * N[(N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(1.0 + N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.9999999999999997e66

   1. Initial program 3.2%

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

   3. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 22.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 23.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 23.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\color{blue}{{\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right) \cdot 2}\right)\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{{\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)} \cdot \color{blue}{\sqrt{2}}\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left({\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(\left({\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   9. Applied egg-rr 25.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot b\right) \cdot \sqrt{2}\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, b\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 26.0%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), \color{blue}{1} \cdot b\right) \cdot \sqrt{2}\right) \]

   if 8.9999999999999997e66 < x-scale

   1. Initial program 2.3%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 60.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   7. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

   9. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \left({angle}^{2}\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot angle\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

       12. PI-lowering-PI.f64 59.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right), 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]

   12. Simplified 59.3%

       \[\leadsto \left(\sqrt{16 \cdot \left(\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right)} + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 9 \cdot 10^{+66}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(a \cdot a\right) \cdot \left(\left(1 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot 0.5\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 25.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{+31}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(b \cdot \sqrt{1 + \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+183}:\\ \;\;\;\;x-scale\_m \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot -0.5\right) + \left(a \cdot a\right) \cdot \left(0.5 \cdot \left(1 + 1\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 1.6e+31)
   (*
    (* 0.25 (* y-scale_m (sqrt 8.0)))
    (* b (sqrt (+ 1.0 (cos (* (* PI angle) 0.011111111111111112))))))
   (if (<= a 1.1e+183)
     (*
      x-scale_m
      (*
       0.25
       (sqrt
        (*
         16.0
         (+
          (*
           (* b b)
           (+ 0.5 (* (cos (* PI (* angle 0.011111111111111112))) -0.5)))
          (* (* a a) (* 0.5 (+ 1.0 1.0))))))))
     (* x-scale_m a))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.6e+31) {
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (b * sqrt((1.0 + cos(((((double) M_PI) * angle) * 0.011111111111111112)))));
	} else if (a <= 1.1e+183) {
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((b * b) * (0.5 + (cos((((double) M_PI) * (angle * 0.011111111111111112))) * -0.5))) + ((a * a) * (0.5 * (1.0 + 1.0)))))));
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.6e+31) {
		tmp = (0.25 * (y_45_scale_m * Math.sqrt(8.0))) * (b * Math.sqrt((1.0 + Math.cos(((Math.PI * angle) * 0.011111111111111112)))));
	} else if (a <= 1.1e+183) {
		tmp = x_45_scale_m * (0.25 * Math.sqrt((16.0 * (((b * b) * (0.5 + (Math.cos((Math.PI * (angle * 0.011111111111111112))) * -0.5))) + ((a * a) * (0.5 * (1.0 + 1.0)))))));
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 1.6e+31:
		tmp = (0.25 * (y_45_scale_m * math.sqrt(8.0))) * (b * math.sqrt((1.0 + math.cos(((math.pi * angle) * 0.011111111111111112)))))
	elif a <= 1.1e+183:
		tmp = x_45_scale_m * (0.25 * math.sqrt((16.0 * (((b * b) * (0.5 + (math.cos((math.pi * (angle * 0.011111111111111112))) * -0.5))) + ((a * a) * (0.5 * (1.0 + 1.0)))))))
	else:
		tmp = x_45_scale_m * a
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 1.6e+31)
		tmp = Float64(Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0))) * Float64(b * sqrt(Float64(1.0 + cos(Float64(Float64(pi * angle) * 0.011111111111111112))))));
	elseif (a <= 1.1e+183)
		tmp = Float64(x_45_scale_m * Float64(0.25 * sqrt(Float64(16.0 * Float64(Float64(Float64(b * b) * Float64(0.5 + Float64(cos(Float64(pi * Float64(angle * 0.011111111111111112))) * -0.5))) + Float64(Float64(a * a) * Float64(0.5 * Float64(1.0 + 1.0))))))));
	else
		tmp = Float64(x_45_scale_m * a);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 1.6e+31)
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (b * sqrt((1.0 + cos(((pi * angle) * 0.011111111111111112)))));
	elseif (a <= 1.1e+183)
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((b * b) * (0.5 + (cos((pi * (angle * 0.011111111111111112))) * -0.5))) + ((a * a) * (0.5 * (1.0 + 1.0)))))));
	else
		tmp = x_45_scale_m * a;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 1.6e+31], N[(N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * N[Sqrt[N[(1.0 + N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.1e+183], N[(x$45$scale$95$m * N[(0.25 * N[Sqrt[N[(16.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$45$scale$95$m * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.6e31

   1. Initial program 2.9%

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

   3. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 22.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\left(a \cdot a\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left({\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}^{2}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{180}{angle}\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 23.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), 2\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 23.3%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\color{blue}{{\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)} \]

   9. Applied egg-rr 18.6%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{e^{\log \left(2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a \cdot a\right) + \left(b \cdot b\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)\right) \cdot 0.5}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \color{blue}{\left(b \cdot \sqrt{1 + \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(\sqrt{1 + \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{sqrt.f64}\left(\left(1 + \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{cos.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{cos.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right)\right) \]

       8. PI-lowering-PI.f64 23.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right)\right) \]

   12. Simplified 23.6%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(b \cdot \sqrt{1 + \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\right)} \]

   if 1.6e31 < a < 1.09999999999999995e183

   1. Initial program 5.6%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 15.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   7. Applied egg-rr 15.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

   9. Applied egg-rr 15.9%

      \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]
   11. Step-by-step derivation

   12. Simplified 15.9%

       \[\leadsto \left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\color{blue}{1} + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale \]

   if 1.09999999999999995e183 < a

   1. Initial program 0.2%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 18.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   7. Applied egg-rr 13.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

   9. Applied egg-rr 13.9%

      \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{a \cdot x-scale} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 22.8%

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

   12. Simplified 22.8%

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

4. Final simplification 22.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{+31}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(b \cdot \sqrt{1 + \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)}\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+183}:\\ \;\;\;\;x-scale \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot -0.5\right) + \left(a \cdot a\right) \cdot \left(0.5 \cdot \left(1 + 1\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 25.9% accurate, 11.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+30}:\\ \;\;\;\;\left(y-scale\_m \cdot \left(0.25 \cdot b\right)\right) \cdot 4\\ \mathbf{elif}\;a \leq 1.42 \cdot 10^{+182}:\\ \;\;\;\;x-scale\_m \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot -0.5\right) + \left(a \cdot a\right) \cdot \left(0.5 \cdot \left(1 + 1\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 7.5e+30)
   (* (* y-scale_m (* 0.25 b)) 4.0)
   (if (<= a 1.42e+182)
     (*
      x-scale_m
      (*
       0.25
       (sqrt
        (*
         16.0
         (+
          (*
           (* b b)
           (+ 0.5 (* (cos (* PI (* angle 0.011111111111111112))) -0.5)))
          (* (* a a) (* 0.5 (+ 1.0 1.0))))))))
     (* x-scale_m a))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 7.5e+30) {
		tmp = (y_45_scale_m * (0.25 * b)) * 4.0;
	} else if (a <= 1.42e+182) {
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((b * b) * (0.5 + (cos((((double) M_PI) * (angle * 0.011111111111111112))) * -0.5))) + ((a * a) * (0.5 * (1.0 + 1.0)))))));
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 7.5e+30) {
		tmp = (y_45_scale_m * (0.25 * b)) * 4.0;
	} else if (a <= 1.42e+182) {
		tmp = x_45_scale_m * (0.25 * Math.sqrt((16.0 * (((b * b) * (0.5 + (Math.cos((Math.PI * (angle * 0.011111111111111112))) * -0.5))) + ((a * a) * (0.5 * (1.0 + 1.0)))))));
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 7.5e+30:
		tmp = (y_45_scale_m * (0.25 * b)) * 4.0
	elif a <= 1.42e+182:
		tmp = x_45_scale_m * (0.25 * math.sqrt((16.0 * (((b * b) * (0.5 + (math.cos((math.pi * (angle * 0.011111111111111112))) * -0.5))) + ((a * a) * (0.5 * (1.0 + 1.0)))))))
	else:
		tmp = x_45_scale_m * a
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 7.5e+30)
		tmp = Float64(Float64(y_45_scale_m * Float64(0.25 * b)) * 4.0);
	elseif (a <= 1.42e+182)
		tmp = Float64(x_45_scale_m * Float64(0.25 * sqrt(Float64(16.0 * Float64(Float64(Float64(b * b) * Float64(0.5 + Float64(cos(Float64(pi * Float64(angle * 0.011111111111111112))) * -0.5))) + Float64(Float64(a * a) * Float64(0.5 * Float64(1.0 + 1.0))))))));
	else
		tmp = Float64(x_45_scale_m * a);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 7.5e+30)
		tmp = (y_45_scale_m * (0.25 * b)) * 4.0;
	elseif (a <= 1.42e+182)
		tmp = x_45_scale_m * (0.25 * sqrt((16.0 * (((b * b) * (0.5 + (cos((pi * (angle * 0.011111111111111112))) * -0.5))) + ((a * a) * (0.5 * (1.0 + 1.0)))))));
	else
		tmp = x_45_scale_m * a;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 7.5e+30], N[(N[(y$45$scale$95$m * N[(0.25 * b), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision], If[LessEqual[a, 1.42e+182], N[(x$45$scale$95$m * N[(0.25 * N[Sqrt[N[(16.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$45$scale$95$m * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 7.49999999999999973e30

   1. Initial program 2.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot b\right), \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\color{blue}{y-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 23.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

   5. Simplified 23.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{2 \cdot 8}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{16}\right) \]

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 23.5%

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

   7. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]

   if 7.49999999999999973e30 < a < 1.41999999999999999e182

   1. Initial program 5.6%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 15.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   7. Applied egg-rr 15.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

   9. Applied egg-rr 15.9%

      \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(16, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, 1\right), \frac{1}{2}\right)\right)\right)\right)\right), \frac{1}{4}\right), x-scale\right) \]
   11. Step-by-step derivation

   12. Simplified 15.9%

       \[\leadsto \left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\color{blue}{1} + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale \]

   if 1.41999999999999999e182 < a

   1. Initial program 0.2%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 18.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   7. Applied egg-rr 13.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

   9. Applied egg-rr 13.9%

      \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{a \cdot x-scale} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 22.8%

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

   12. Simplified 22.8%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+30}:\\ \;\;\;\;\left(y-scale \cdot \left(0.25 \cdot b\right)\right) \cdot 4\\ \mathbf{elif}\;a \leq 1.42 \cdot 10^{+182}:\\ \;\;\;\;x-scale \cdot \left(0.25 \cdot \sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot -0.5\right) + \left(a \cdot a\right) \cdot \left(0.5 \cdot \left(1 + 1\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 25.8% accurate, 229.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+31}:\\ \;\;\;\;\left(y-scale\_m \cdot \left(0.25 \cdot b\right)\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 1.4e+31) (* (* y-scale_m (* 0.25 b)) 4.0) (* x-scale_m a)))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.4e+31) {
		tmp = (y_45_scale_m * (0.25 * b)) * 4.0;
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 1.4d+31) then
        tmp = (y_45scale_m * (0.25d0 * b)) * 4.0d0
    else
        tmp = x_45scale_m * a
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.4e+31) {
		tmp = (y_45_scale_m * (0.25 * b)) * 4.0;
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 1.4e+31:
		tmp = (y_45_scale_m * (0.25 * b)) * 4.0
	else:
		tmp = x_45_scale_m * a
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 1.4e+31)
		tmp = Float64(Float64(y_45_scale_m * Float64(0.25 * b)) * 4.0);
	else
		tmp = Float64(x_45_scale_m * a);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 1.4e+31)
		tmp = (y_45_scale_m * (0.25 * b)) * 4.0;
	else
		tmp = x_45_scale_m * a;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 1.4e+31], N[(N[(y$45$scale$95$m * N[(0.25 * b), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision], N[(x$45$scale$95$m * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.40000000000000008e31

   1. Initial program 2.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot b\right), \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\color{blue}{y-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 23.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

   5. Simplified 23.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{2 \cdot 8}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{16}\right) \]

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 23.5%

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

   7. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]

   if 1.40000000000000008e31 < a

   1. Initial program 3.5%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 16.7%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   7. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

   9. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{a \cdot x-scale} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 14.3%

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

   12. Simplified 14.3%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+31}:\\ \;\;\;\;\left(y-scale \cdot \left(0.25 \cdot b\right)\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 25.8% accurate, 344.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.7 \cdot 10^{+31}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 1.7e+31) (* y-scale_m b) (* x-scale_m a)))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.7e+31) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 1.7d+31) then
        tmp = y_45scale_m * b
    else
        tmp = x_45scale_m * a
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.7e+31) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 1.7e+31:
		tmp = y_45_scale_m * b
	else:
		tmp = x_45_scale_m * a
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 1.7e+31)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(x_45_scale_m * a);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 1.7e+31)
		tmp = y_45_scale_m * b;
	else
		tmp = x_45_scale_m * a;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 1.7e+31], N[(y$45$scale$95$m * b), $MachinePrecision], N[(x$45$scale$95$m * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.6999999999999999e31

   1. Initial program 2.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot b\right), \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\color{blue}{y-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 23.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

   5. Simplified 23.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{2 \cdot 8}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \left(y-scale \cdot \sqrt{16}\right) \]

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 23.5%

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

   7. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot b\right) \cdot y-scale\right) \cdot 4} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot y-scale} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y-scale \cdot \color{blue}{b} \]

      2. *-lowering-*.f64 23.5%

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

   10. Simplified 23.5%

       \[\leadsto \color{blue}{y-scale \cdot b} \]

   if 1.6999999999999999e31 < a

   1. Initial program 3.5%

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

   3. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 16.7%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

   7. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

   9. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{a \cdot x-scale} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 14.3%

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

   12. Simplified 14.3%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.7 \cdot 10^{+31}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 17.4% accurate, 919.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ x-scale\_m \cdot a \end{array} \]

FPCore

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

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	return x_45_scale_m * a;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = x_45scale_m * a
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	return x_45_scale_m * a;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	return x_45_scale_m * a

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	return Float64(x_45_scale_m * a)
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = x_45_scale_m * a;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(x$45$scale$95$m * a), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in y-scale around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(x-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(x-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   7. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

5. Simplified 19.3%

   \[\leadsto \color{blue}{\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

7. Applied egg-rr 16.9%

   \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\frac{2 \cdot \pi}{\frac{180}{angle}}\right)\right)\right)}\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{4} \cdot x-scale\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{x-scale} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{8} \cdot \sqrt{2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)}\right) \cdot \frac{1}{4}\right), \color{blue}{x-scale}\right) \]

9. Applied egg-rr 16.9%

   \[\leadsto \color{blue}{\left(\sqrt{16 \cdot \left(\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) + \left(a \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) + 1\right) \cdot 0.5\right)\right)} \cdot 0.25\right) \cdot x-scale} \]

10. Taylor expanded in angle around 0

    \[\leadsto \color{blue}{a \cdot x-scale} \]
11. Step-by-step derivation

    1. *-lowering-*.f64 14.9%

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

12. Simplified 14.9%

    \[\leadsto \color{blue}{a \cdot x-scale} \]

13. Final simplification 14.9%

    \[\leadsto x-scale \cdot a \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (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)) (sqrt (+ (pow (- (/ (/ (+ (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)) 2.0) (pow (/ (/ (* (* (* 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))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))