a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 60.1%

Time: 40.6s

Alternatives: 17

Speedup: 919.0×

• 31.0% 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: 2.9% 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: 60.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale\_m \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\sqrt{8} \cdot y-scale\_m\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(b \cdot \cos t\_0, a \cdot \sin t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right), a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \end{array} \]

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 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale_m 5e+27)
     (*
      (* (* 0.25 (* (sqrt 8.0) y-scale_m)) (sqrt 2.0))
      (hypot (* b (cos t_0)) (* a (sin t_0))))
     (*
      (sqrt 2.0)
      (*
       (hypot
        (* b (sin (* angle (* 0.005555555555555556 (pow (sqrt PI) 2.0)))))
        (* a (cos (* angle (* 0.005555555555555556 PI)))))
       (* 0.25 (* x-scale_m (sqrt 8.0))))))))

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 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale_m <= 5e+27) {
		tmp = ((0.25 * (sqrt(8.0) * y_45_scale_m)) * sqrt(2.0)) * hypot((b * cos(t_0)), (a * sin(t_0)));
	} else {
		tmp = sqrt(2.0) * (hypot((b * sin((angle * (0.005555555555555556 * pow(sqrt(((double) M_PI)), 2.0))))), (a * cos((angle * (0.005555555555555556 * ((double) M_PI)))))) * (0.25 * (x_45_scale_m * sqrt(8.0))));
	}
	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 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale_m <= 5e+27) {
		tmp = ((0.25 * (Math.sqrt(8.0) * y_45_scale_m)) * Math.sqrt(2.0)) * Math.hypot((b * Math.cos(t_0)), (a * Math.sin(t_0)));
	} else {
		tmp = Math.sqrt(2.0) * (Math.hypot((b * Math.sin((angle * (0.005555555555555556 * Math.pow(Math.sqrt(Math.PI), 2.0))))), (a * Math.cos((angle * (0.005555555555555556 * Math.PI))))) * (0.25 * (x_45_scale_m * Math.sqrt(8.0))));
	}
	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 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale_m <= 5e+27:
		tmp = ((0.25 * (math.sqrt(8.0) * y_45_scale_m)) * math.sqrt(2.0)) * math.hypot((b * math.cos(t_0)), (a * math.sin(t_0)))
	else:
		tmp = math.sqrt(2.0) * (math.hypot((b * math.sin((angle * (0.005555555555555556 * math.pow(math.sqrt(math.pi), 2.0))))), (a * math.cos((angle * (0.005555555555555556 * math.pi))))) * (0.25 * (x_45_scale_m * math.sqrt(8.0))))
	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(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale_m <= 5e+27)
		tmp = Float64(Float64(Float64(0.25 * Float64(sqrt(8.0) * y_45_scale_m)) * sqrt(2.0)) * hypot(Float64(b * cos(t_0)), Float64(a * sin(t_0))));
	else
		tmp = Float64(sqrt(2.0) * Float64(hypot(Float64(b * sin(Float64(angle * Float64(0.005555555555555556 * (sqrt(pi) ^ 2.0))))), Float64(a * cos(Float64(angle * Float64(0.005555555555555556 * pi))))) * Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0)))));
	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 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale_m <= 5e+27)
		tmp = ((0.25 * (sqrt(8.0) * y_45_scale_m)) * sqrt(2.0)) * hypot((b * cos(t_0)), (a * sin(t_0)));
	else
		tmp = sqrt(2.0) * (hypot((b * sin((angle * (0.005555555555555556 * (sqrt(pi) ^ 2.0))))), (a * cos((angle * (0.005555555555555556 * pi))))) * (0.25 * (x_45_scale_m * sqrt(8.0))));
	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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 5e+27], N[(N[(N[(0.25 * N[(N[Sqrt[8.0], $MachinePrecision] * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(b * N[Sin[N[(angle * N[(0.005555555555555556 * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a * N[Cos[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.99999999999999979e27

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 21.9%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

   7. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left(\sqrt{8} \cdot y-scale\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   if 4.99999999999999979e27 < x-scale

   1. Initial program 3.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 63.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt N/A

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

      2. pow2 N/A

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

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

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

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

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

      5. PI-lowering-PI.f64 68.2%

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

   9. Applied egg-rr 68.2%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\sqrt{8} \cdot y-scale\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right), a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale\_m \leq 4.5 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\sqrt{8} \cdot y-scale\_m\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(b \cdot \cos t\_0, a \cdot \sin t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale\_m \cdot {\left({8}^{0.25}\right)}^{2}\right)\right) \cdot \mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), a\right)\right)\\ \end{array} \end{array} \]

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 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale_m 4.5e+27)
     (*
      (* (* 0.25 (* (sqrt 8.0) y-scale_m)) (sqrt 2.0))
      (hypot (* b (cos t_0)) (* a (sin t_0))))
     (*
      (sqrt 2.0)
      (*
       (* 0.25 (* x-scale_m (pow (pow 8.0 0.25) 2.0)))
       (hypot (* b (sin (* angle (* 0.005555555555555556 PI)))) 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 t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale_m <= 4.5e+27) {
		tmp = ((0.25 * (sqrt(8.0) * y_45_scale_m)) * sqrt(2.0)) * hypot((b * cos(t_0)), (a * sin(t_0)));
	} else {
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * pow(pow(8.0, 0.25), 2.0))) * hypot((b * sin((angle * (0.005555555555555556 * ((double) M_PI))))), 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 t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale_m <= 4.5e+27) {
		tmp = ((0.25 * (Math.sqrt(8.0) * y_45_scale_m)) * Math.sqrt(2.0)) * Math.hypot((b * Math.cos(t_0)), (a * Math.sin(t_0)));
	} else {
		tmp = Math.sqrt(2.0) * ((0.25 * (x_45_scale_m * Math.pow(Math.pow(8.0, 0.25), 2.0))) * Math.hypot((b * Math.sin((angle * (0.005555555555555556 * Math.PI)))), 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):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale_m <= 4.5e+27:
		tmp = ((0.25 * (math.sqrt(8.0) * y_45_scale_m)) * math.sqrt(2.0)) * math.hypot((b * math.cos(t_0)), (a * math.sin(t_0)))
	else:
		tmp = math.sqrt(2.0) * ((0.25 * (x_45_scale_m * math.pow(math.pow(8.0, 0.25), 2.0))) * math.hypot((b * math.sin((angle * (0.005555555555555556 * math.pi)))), 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)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale_m <= 4.5e+27)
		tmp = Float64(Float64(Float64(0.25 * Float64(sqrt(8.0) * y_45_scale_m)) * sqrt(2.0)) * hypot(Float64(b * cos(t_0)), Float64(a * sin(t_0))));
	else
		tmp = Float64(sqrt(2.0) * Float64(Float64(0.25 * Float64(x_45_scale_m * ((8.0 ^ 0.25) ^ 2.0))) * hypot(Float64(b * sin(Float64(angle * Float64(0.005555555555555556 * pi)))), 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)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale_m <= 4.5e+27)
		tmp = ((0.25 * (sqrt(8.0) * y_45_scale_m)) * sqrt(2.0)) * hypot((b * cos(t_0)), (a * sin(t_0)));
	else
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * ((8.0 ^ 0.25) ^ 2.0))) * hypot((b * sin((angle * (0.005555555555555556 * pi)))), 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_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 4.5e+27], N[(N[(N[(0.25 * N[(N[Sqrt[8.0], $MachinePrecision] * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.25 * N[(x$45$scale$95$m * N[Power[N[Power[8.0, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(b * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + a ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.4999999999999999e27

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 21.9%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

   7. Applied egg-rr 25.8%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left(\sqrt{8} \cdot y-scale\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   if 4.4999999999999999e27 < x-scale

   1. Initial program 3.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 63.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 68.1%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 68.3%

       \[\leadsto \sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \color{blue}{1}\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]
   11. Step-by-step derivation

       1. pow1/2 N/A

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

       2. sqr-pow N/A

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

       3. pow2 N/A

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

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

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

       5. metadata-eval N/A

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

       6. pow-lowering-pow.f64 68.4%

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

   12. Applied egg-rr 68.4%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 4.5 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\sqrt{8} \cdot y-scale\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot {\left({8}^{0.25}\right)}^{2}\right)\right) \cdot \mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;y-scale\_m \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \left(0.25 \cdot \mathsf{hypot}\left(b \cdot \sin t\_0, a \cdot \cos t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale\_m \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

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 (* angle (* 0.005555555555555556 PI))))
   (if (<= y-scale_m 7.2e+59)
     (*
      (* x-scale_m (sqrt 8.0))
      (* (sqrt 2.0) (* 0.25 (hypot (* b (sin t_0)) (* a (cos t_0))))))
     (*
      (* 0.25 b)
      (*
       (*
        y-scale_m
        (+
         1.0
         (*
          (* angle angle)
          (+
           (* (* PI PI) -1.54320987654321e-5)
           (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0))))))
       (* (sqrt 8.0) (sqrt 2.0)))))))

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 = angle * (0.005555555555555556 * ((double) M_PI));
	double tmp;
	if (y_45_scale_m <= 7.2e+59) {
		tmp = (x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * (0.25 * hypot((b * sin(t_0)), (a * cos(t_0)))));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	}
	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 = angle * (0.005555555555555556 * Math.PI);
	double tmp;
	if (y_45_scale_m <= 7.2e+59) {
		tmp = (x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * (0.25 * Math.hypot((b * Math.sin(t_0)), (a * Math.cos(t_0)))));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0)))))) * (Math.sqrt(8.0) * Math.sqrt(2.0)));
	}
	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 = angle * (0.005555555555555556 * math.pi)
	tmp = 0
	if y_45_scale_m <= 7.2e+59:
		tmp = (x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * (0.25 * math.hypot((b * math.sin(t_0)), (a * math.cos(t_0)))))
	else:
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0)))))) * (math.sqrt(8.0) * math.sqrt(2.0)))
	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(angle * Float64(0.005555555555555556 * pi))
	tmp = 0.0
	if (y_45_scale_m <= 7.2e+59)
		tmp = Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * Float64(0.25 * hypot(Float64(b * sin(t_0)), Float64(a * cos(t_0))))));
	else
		tmp = Float64(Float64(0.25 * b) * Float64(Float64(y_45_scale_m * Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * Float64(sqrt(8.0) * sqrt(2.0))));
	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 = angle * (0.005555555555555556 * pi);
	tmp = 0.0;
	if (y_45_scale_m <= 7.2e+59)
		tmp = (x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * (0.25 * hypot((b * sin(t_0)), (a * cos(t_0)))));
	else
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	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[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 7.2e+59], N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 * N[Sqrt[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * b), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.1999999999999997e59

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 23.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 25.2%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

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

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

   9. Applied egg-rr 25.1%

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

   if 7.1999999999999997e59 < y-scale

   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 b around inf

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

   5. Simplified 18.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{\frac{4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} + {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}} \]

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 28.7%

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

   8. Simplified 28.7%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}, y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{2}\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 26.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   11. Simplified 26.0%

       \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(\color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)} \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \left(0.25 \cdot \mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;y-scale\_m \leq 1.95 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(b \cdot \sin t\_0, a \cdot \cos t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale\_m \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

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 (* angle (* 0.005555555555555556 PI))))
   (if (<= y-scale_m 1.95e+59)
     (*
      (sqrt 2.0)
      (*
       (* 0.25 (* x-scale_m (sqrt 8.0)))
       (hypot (* b (sin t_0)) (* a (cos t_0)))))
     (*
      (* 0.25 b)
      (*
       (*
        y-scale_m
        (+
         1.0
         (*
          (* angle angle)
          (+
           (* (* PI PI) -1.54320987654321e-5)
           (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0))))))
       (* (sqrt 8.0) (sqrt 2.0)))))))

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 = angle * (0.005555555555555556 * ((double) M_PI));
	double tmp;
	if (y_45_scale_m <= 1.95e+59) {
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((b * sin(t_0)), (a * cos(t_0))));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	}
	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 = angle * (0.005555555555555556 * Math.PI);
	double tmp;
	if (y_45_scale_m <= 1.95e+59) {
		tmp = Math.sqrt(2.0) * ((0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.hypot((b * Math.sin(t_0)), (a * Math.cos(t_0))));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0)))))) * (Math.sqrt(8.0) * Math.sqrt(2.0)));
	}
	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 = angle * (0.005555555555555556 * math.pi)
	tmp = 0
	if y_45_scale_m <= 1.95e+59:
		tmp = math.sqrt(2.0) * ((0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.hypot((b * math.sin(t_0)), (a * math.cos(t_0))))
	else:
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0)))))) * (math.sqrt(8.0) * math.sqrt(2.0)))
	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(angle * Float64(0.005555555555555556 * pi))
	tmp = 0.0
	if (y_45_scale_m <= 1.95e+59)
		tmp = Float64(sqrt(2.0) * Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * hypot(Float64(b * sin(t_0)), Float64(a * cos(t_0)))));
	else
		tmp = Float64(Float64(0.25 * b) * Float64(Float64(y_45_scale_m * Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * Float64(sqrt(8.0) * sqrt(2.0))));
	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 = angle * (0.005555555555555556 * pi);
	tmp = 0.0;
	if (y_45_scale_m <= 1.95e+59)
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((b * sin(t_0)), (a * cos(t_0))));
	else
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	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[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 1.95e+59], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * b), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.95000000000000011e59

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 23.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 25.2%

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

   if 1.95000000000000011e59 < y-scale

   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 b around inf

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

   5. Simplified 18.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{\frac{4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} + {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}} \]

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 28.7%

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

   8. Simplified 28.7%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}, y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{2}\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 26.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   11. Simplified 26.0%

       \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(\color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)} \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.95 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale\_m \cdot {\left({8}^{0.25}\right)}^{2}\right)\right) \cdot \mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale\_m \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

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 (<= y-scale_m 1.15e+63)
   (*
    (sqrt 2.0)
    (*
     (* 0.25 (* x-scale_m (pow (pow 8.0 0.25) 2.0)))
     (hypot (* b (sin (* angle (* 0.005555555555555556 PI)))) a)))
   (*
    (* 0.25 b)
    (*
     (*
      y-scale_m
      (+
       1.0
       (*
        (* angle angle)
        (+
         (* (* PI PI) -1.54320987654321e-5)
         (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0))))))
     (* (sqrt 8.0) (sqrt 2.0))))))

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 (y_45_scale_m <= 1.15e+63) {
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * pow(pow(8.0, 0.25), 2.0))) * hypot((b * sin((angle * (0.005555555555555556 * ((double) M_PI))))), a));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	}
	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 (y_45_scale_m <= 1.15e+63) {
		tmp = Math.sqrt(2.0) * ((0.25 * (x_45_scale_m * Math.pow(Math.pow(8.0, 0.25), 2.0))) * Math.hypot((b * Math.sin((angle * (0.005555555555555556 * Math.PI)))), a));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0)))))) * (Math.sqrt(8.0) * Math.sqrt(2.0)));
	}
	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 y_45_scale_m <= 1.15e+63:
		tmp = math.sqrt(2.0) * ((0.25 * (x_45_scale_m * math.pow(math.pow(8.0, 0.25), 2.0))) * math.hypot((b * math.sin((angle * (0.005555555555555556 * math.pi)))), a))
	else:
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0)))))) * (math.sqrt(8.0) * math.sqrt(2.0)))
	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 (y_45_scale_m <= 1.15e+63)
		tmp = Float64(sqrt(2.0) * Float64(Float64(0.25 * Float64(x_45_scale_m * ((8.0 ^ 0.25) ^ 2.0))) * hypot(Float64(b * sin(Float64(angle * Float64(0.005555555555555556 * pi)))), a)));
	else
		tmp = Float64(Float64(0.25 * b) * Float64(Float64(y_45_scale_m * Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * Float64(sqrt(8.0) * sqrt(2.0))));
	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 (y_45_scale_m <= 1.15e+63)
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * ((8.0 ^ 0.25) ^ 2.0))) * hypot((b * sin((angle * (0.005555555555555556 * pi)))), a));
	else
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	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[y$45$scale$95$m, 1.15e+63], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.25 * N[(x$45$scale$95$m * N[Power[N[Power[8.0, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(b * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + a ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * b), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.14999999999999997e63

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 23.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 25.2%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 25.1%

       \[\leadsto \sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \color{blue}{1}\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]
   11. Step-by-step derivation

       1. pow1/2 N/A

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

       2. sqr-pow N/A

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

       3. pow2 N/A

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

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

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

       5. metadata-eval N/A

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

       6. pow-lowering-pow.f64 25.2%

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

   12. Applied egg-rr 25.2%

       \[\leadsto \sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot 1\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \color{blue}{{\left({8}^{0.25}\right)}^{2}}\right)\right)\right) \]

   if 1.14999999999999997e63 < y-scale

   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 b around inf

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

   5. Simplified 18.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{\frac{4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} + {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}} \]

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 28.7%

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

   8. Simplified 28.7%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}, y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{2}\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 26.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   11. Simplified 26.0%

       \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(\color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)} \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot {\left({8}^{0.25}\right)}^{2}\right)\right) \cdot \mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.7 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale\_m \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

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 (<= y-scale_m 2.7e+58)
   (*
    (sqrt 2.0)
    (*
     (* 0.25 (* x-scale_m (sqrt 8.0)))
     (hypot (* b (sin (* angle (* 0.005555555555555556 PI)))) a)))
   (*
    (* 0.25 b)
    (*
     (*
      y-scale_m
      (+
       1.0
       (*
        (* angle angle)
        (+
         (* (* PI PI) -1.54320987654321e-5)
         (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0))))))
     (* (sqrt 8.0) (sqrt 2.0))))))

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 (y_45_scale_m <= 2.7e+58) {
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((b * sin((angle * (0.005555555555555556 * ((double) M_PI))))), a));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	}
	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 (y_45_scale_m <= 2.7e+58) {
		tmp = Math.sqrt(2.0) * ((0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.hypot((b * Math.sin((angle * (0.005555555555555556 * Math.PI)))), a));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0)))))) * (Math.sqrt(8.0) * Math.sqrt(2.0)));
	}
	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 y_45_scale_m <= 2.7e+58:
		tmp = math.sqrt(2.0) * ((0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.hypot((b * math.sin((angle * (0.005555555555555556 * math.pi)))), a))
	else:
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0)))))) * (math.sqrt(8.0) * math.sqrt(2.0)))
	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 (y_45_scale_m <= 2.7e+58)
		tmp = Float64(sqrt(2.0) * Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * hypot(Float64(b * sin(Float64(angle * Float64(0.005555555555555556 * pi)))), a)));
	else
		tmp = Float64(Float64(0.25 * b) * Float64(Float64(y_45_scale_m * Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * Float64(sqrt(8.0) * sqrt(2.0))));
	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 (y_45_scale_m <= 2.7e+58)
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((b * sin((angle * (0.005555555555555556 * pi)))), a));
	else
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	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[y$45$scale$95$m, 2.7e+58], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(b * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + a ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * b), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.7000000000000001e58

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 23.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 25.2%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 25.1%

       \[\leadsto \sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \color{blue}{1}\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-rgt-identity 25.1%

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

   12. Applied egg-rr 25.1%

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

   if 2.7000000000000001e58 < y-scale

   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 b around inf

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

   5. Simplified 18.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{\frac{4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} + {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}} \]

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 28.7%

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

   8. Simplified 28.7%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}, y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{2}\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 26.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   11. Simplified 26.0%

       \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(\color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)} \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.7 \cdot 10^{+58}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 10^{+62}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{8} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), a\right) \cdot \left(x-scale\_m \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale\_m \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

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 (<= y-scale_m 1e+62)
   (*
    (sqrt 2.0)
    (*
     (sqrt 8.0)
     (*
      (hypot (* b (sin (* 0.005555555555555556 (* angle PI)))) a)
      (* x-scale_m 0.25))))
   (*
    (* 0.25 b)
    (*
     (*
      y-scale_m
      (+
       1.0
       (*
        (* angle angle)
        (+
         (* (* PI PI) -1.54320987654321e-5)
         (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0))))))
     (* (sqrt 8.0) (sqrt 2.0))))))

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 (y_45_scale_m <= 1e+62) {
		tmp = sqrt(2.0) * (sqrt(8.0) * (hypot((b * sin((0.005555555555555556 * (angle * ((double) M_PI))))), a) * (x_45_scale_m * 0.25)));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	}
	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 (y_45_scale_m <= 1e+62) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(8.0) * (Math.hypot((b * Math.sin((0.005555555555555556 * (angle * Math.PI)))), a) * (x_45_scale_m * 0.25)));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0)))))) * (Math.sqrt(8.0) * Math.sqrt(2.0)));
	}
	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 y_45_scale_m <= 1e+62:
		tmp = math.sqrt(2.0) * (math.sqrt(8.0) * (math.hypot((b * math.sin((0.005555555555555556 * (angle * math.pi)))), a) * (x_45_scale_m * 0.25)))
	else:
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0)))))) * (math.sqrt(8.0) * math.sqrt(2.0)))
	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 (y_45_scale_m <= 1e+62)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(8.0) * Float64(hypot(Float64(b * sin(Float64(0.005555555555555556 * Float64(angle * pi)))), a) * Float64(x_45_scale_m * 0.25))));
	else
		tmp = Float64(Float64(0.25 * b) * Float64(Float64(y_45_scale_m * Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * Float64(sqrt(8.0) * sqrt(2.0))));
	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 (y_45_scale_m <= 1e+62)
		tmp = sqrt(2.0) * (sqrt(8.0) * (hypot((b * sin((0.005555555555555556 * (angle * pi)))), a) * (x_45_scale_m * 0.25)));
	else
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	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[y$45$scale$95$m, 1e+62], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Sqrt[N[(b * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + a ^ 2], $MachinePrecision] * N[(x$45$scale$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * b), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.00000000000000004e62

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 23.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 25.2%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 25.1%

       \[\leadsto \sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \color{blue}{1}\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

   12. Applied egg-rr 25.1%

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

   if 1.00000000000000004e62 < y-scale

   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 b around inf

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

   5. Simplified 18.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{\frac{4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} + {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}} \]

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 28.7%

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

   8. Simplified 28.7%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}, y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{2}\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 26.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   11. Simplified 26.0%

       \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(\color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)} \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 10^{+62}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{8} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), a\right) \cdot \left(x-scale \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 6.6 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale\_m \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

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 (<= y-scale_m 6.6e+62)
   (*
    (sqrt 2.0)
    (*
     (* 0.25 (* x-scale_m (sqrt 8.0)))
     (hypot (* 0.005555555555555556 (* angle (* b PI))) a)))
   (*
    (* 0.25 b)
    (*
     (*
      y-scale_m
      (+
       1.0
       (*
        (* angle angle)
        (+
         (* (* PI PI) -1.54320987654321e-5)
         (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0))))))
     (* (sqrt 8.0) (sqrt 2.0))))))

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 (y_45_scale_m <= 6.6e+62) {
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((0.005555555555555556 * (angle * (b * ((double) M_PI)))), a));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	}
	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 (y_45_scale_m <= 6.6e+62) {
		tmp = Math.sqrt(2.0) * ((0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.hypot((0.005555555555555556 * (angle * (b * Math.PI))), a));
	} else {
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0)))))) * (Math.sqrt(8.0) * Math.sqrt(2.0)));
	}
	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 y_45_scale_m <= 6.6e+62:
		tmp = math.sqrt(2.0) * ((0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.hypot((0.005555555555555556 * (angle * (b * math.pi))), a))
	else:
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0)))))) * (math.sqrt(8.0) * math.sqrt(2.0)))
	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 (y_45_scale_m <= 6.6e+62)
		tmp = Float64(sqrt(2.0) * Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * hypot(Float64(0.005555555555555556 * Float64(angle * Float64(b * pi))), a)));
	else
		tmp = Float64(Float64(0.25 * b) * Float64(Float64(y_45_scale_m * Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * Float64(sqrt(8.0) * sqrt(2.0))));
	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 (y_45_scale_m <= 6.6e+62)
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((0.005555555555555556 * (angle * (b * pi))), a));
	else
		tmp = (0.25 * b) * ((y_45_scale_m * (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))) * (sqrt(8.0) * sqrt(2.0)));
	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[y$45$scale$95$m, 6.6e+62], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.005555555555555556 * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + a ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * b), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 6.6e62

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 23.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 25.2%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 25.1%

       \[\leadsto \sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \color{blue}{1}\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

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

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

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

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

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

       3. *-commutative N/A

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

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

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

       5. PI-lowering-PI.f64 25.8%

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

   13. Simplified 25.8%

       \[\leadsto \sqrt{2} \cdot \left(\mathsf{hypot}\left(\color{blue}{0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)}, a \cdot 1\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]

   if 6.6e62 < y-scale

   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 b around inf

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

   5. Simplified 18.8%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{\frac{4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} + {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}} \]

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 28.7%

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

   8. Simplified 28.7%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}, y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{2}\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 26.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right), y-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   11. Simplified 26.0%

       \[\leadsto \left(0.25 \cdot b\right) \cdot \left(\left(\color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)} \cdot y-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 6.6 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 4.5 \cdot 10^{+27}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right), a\right)\right)\\ \end{array} \end{array} \]

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 4.5e+27)
   (* y-scale_m b)
   (*
    (sqrt 2.0)
    (*
     (* 0.25 (* x-scale_m (sqrt 8.0)))
     (hypot (* 0.005555555555555556 (* angle (* b PI))) 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 (x_45_scale_m <= 4.5e+27) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((0.005555555555555556 * (angle * (b * ((double) M_PI)))), 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 (x_45_scale_m <= 4.5e+27) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = Math.sqrt(2.0) * ((0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.hypot((0.005555555555555556 * (angle * (b * Math.PI))), 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 x_45_scale_m <= 4.5e+27:
		tmp = y_45_scale_m * b
	else:
		tmp = math.sqrt(2.0) * ((0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.hypot((0.005555555555555556 * (angle * (b * math.pi))), 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 (x_45_scale_m <= 4.5e+27)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(sqrt(2.0) * Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * hypot(Float64(0.005555555555555556 * Float64(angle * Float64(b * pi))), 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 (x_45_scale_m <= 4.5e+27)
		tmp = y_45_scale_m * b;
	else
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((0.005555555555555556 * (angle * (b * pi))), 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[x$45$scale$95$m, 4.5e+27], N[(y$45$scale$95$m * b), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.005555555555555556 * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + a ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.4999999999999999e27

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 19.2%

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

   5. Simplified 19.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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 19.3%

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

   7. Applied egg-rr 19.3%

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

   8. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 19.3%

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

   10. Simplified 19.3%

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

   if 4.4999999999999999e27 < x-scale

   1. Initial program 3.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 63.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 68.1%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 68.3%

       \[\leadsto \sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \color{blue}{1}\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

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

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

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

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

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

       3. *-commutative N/A

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

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

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

       5. PI-lowering-PI.f64 66.7%

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

   13. Simplified 66.7%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 4.5 \cdot 10^{+27}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right), a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 24.7% accurate, 12.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 0.00065:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(a + 1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{\pi \cdot \pi}{a}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 0.00065)
   (* y-scale_m b)
   (*
    (sqrt 2.0)
    (*
     (* 0.25 (* x-scale_m (sqrt 8.0)))
     (+
      a
      (*
       1.54320987654321e-5
       (* (* angle angle) (* (* b b) (/ (* PI PI) 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 (x_45_scale_m <= 0.00065) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * (a + (1.54320987654321e-5 * ((angle * angle) * ((b * b) * ((((double) M_PI) * ((double) M_PI)) / 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 (x_45_scale_m <= 0.00065) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = Math.sqrt(2.0) * ((0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (a + (1.54320987654321e-5 * ((angle * angle) * ((b * b) * ((Math.PI * Math.PI) / 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 x_45_scale_m <= 0.00065:
		tmp = y_45_scale_m * b
	else:
		tmp = math.sqrt(2.0) * ((0.25 * (x_45_scale_m * math.sqrt(8.0))) * (a + (1.54320987654321e-5 * ((angle * angle) * ((b * b) * ((math.pi * math.pi) / 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 (x_45_scale_m <= 0.00065)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(sqrt(2.0) * Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(a + Float64(1.54320987654321e-5 * Float64(Float64(angle * angle) * Float64(Float64(b * b) * Float64(Float64(pi * pi) / 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 (x_45_scale_m <= 0.00065)
		tmp = y_45_scale_m * b;
	else
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * (a + (1.54320987654321e-5 * ((angle * angle) * ((b * b) * ((pi * pi) / 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[x$45$scale$95$m, 0.00065], N[(y$45$scale$95$m * b), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + N[(1.54320987654321e-5 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 6.4999999999999997e-4

   1. Initial program 2.5%

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

   3. Taylor expanded in 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 18.9%

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

   5. Simplified 18.9%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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 19.1%

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

   7. Applied egg-rr 19.1%

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

   8. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 19.1%

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

   10. Simplified 19.1%

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

   if 6.4999999999999997e-4 < x-scale

   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 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 58.7%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 62.2%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 62.1%

       \[\leadsto \sqrt{2} \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot \color{blue}{1}\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\color{blue}{\left(a + \frac{1}{64800} \cdot \frac{{angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{a}\right)}, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{1}{64800} \cdot \frac{{angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{a}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{4}}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \left(\frac{{angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{a}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \left({angle}^{2} \cdot \frac{{b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{a}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{{b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{a}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{{b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{a}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{{b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{a}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{a}\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left(\frac{{\mathsf{PI}\left(\right)}^{2}}{a}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{{\mathsf{PI}\left(\right)}^{2}}{a}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{{\mathsf{PI}\left(\right)}^{2}}{a}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{/.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), a\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), a\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), a\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), a\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

       15. PI-lowering-PI.f64 22.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(a, \mathsf{*.f64}\left(\frac{1}{64800}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), a\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]

   13. Simplified 22.7%

       \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\left(a + 1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{\pi \cdot \pi}{a}\right)\right)\right)} \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 0.00065:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(a + 1.54320987654321 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{\pi \cdot \pi}{a}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{+94}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(\sqrt{8} \cdot y-scale\_m\right) \cdot \left(x-scale\_m \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale\_m}\\ \end{array} \end{array} \]

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 (<= b 1e+94)
   (* (* 0.25 (* x-scale_m (sqrt 8.0))) (* (sqrt 2.0) a))
   (*
    (* 0.25 (* (* (sqrt 8.0) y-scale_m) (* x-scale_m b)))
    (/ (sqrt 2.0) x-scale_m))))

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 (b <= 1e+94) {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	} else {
		tmp = (0.25 * ((sqrt(8.0) * y_45_scale_m) * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	}
	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 (b <= 1d+94) then
        tmp = (0.25d0 * (x_45scale_m * sqrt(8.0d0))) * (sqrt(2.0d0) * a)
    else
        tmp = (0.25d0 * ((sqrt(8.0d0) * y_45scale_m) * (x_45scale_m * b))) * (sqrt(2.0d0) / x_45scale_m)
    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 (b <= 1e+94) {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.sqrt(2.0) * a);
	} else {
		tmp = (0.25 * ((Math.sqrt(8.0) * y_45_scale_m) * (x_45_scale_m * b))) * (Math.sqrt(2.0) / x_45_scale_m);
	}
	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 b <= 1e+94:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.sqrt(2.0) * a)
	else:
		tmp = (0.25 * ((math.sqrt(8.0) * y_45_scale_m) * (x_45_scale_m * b))) * (math.sqrt(2.0) / x_45_scale_m)
	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 (b <= 1e+94)
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * a));
	else
		tmp = Float64(Float64(0.25 * Float64(Float64(sqrt(8.0) * y_45_scale_m) * Float64(x_45_scale_m * b))) * Float64(sqrt(2.0) / x_45_scale_m));
	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 (b <= 1e+94)
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	else
		tmp = (0.25 * ((sqrt(8.0) * y_45_scale_m) * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	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[b, 1e+94], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] * N[(x$45$scale$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1e94

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 23.0%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 17.5%

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

   8. Simplified 17.5%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)} \]

   if 1e94 < b

   1. Initial program 2.1%

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

   3. Taylor expanded in b around inf

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

   5. Simplified 18.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\sqrt{\frac{4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)} + {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot x-scale} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}\right)}} \]

   6. Taylor expanded in angle around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 37.7%

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

   8. Simplified 37.7%

      \[\leadsto \left(0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \color{blue}{\frac{\sqrt{2}}{x-scale}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+94}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(\sqrt{8} \cdot y-scale\right) \cdot \left(x-scale \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 23.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 1.35 \cdot 10^{+31}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \end{array} \end{array} \]

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 1.35e+31)
   (* y-scale_m b)
   (* (* 0.25 (* x-scale_m (sqrt 8.0))) (* (sqrt 2.0) 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 (x_45_scale_m <= 1.35e+31) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * 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 (x_45scale_m <= 1.35d+31) then
        tmp = y_45scale_m * b
    else
        tmp = (0.25d0 * (x_45scale_m * sqrt(8.0d0))) * (sqrt(2.0d0) * 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 (x_45_scale_m <= 1.35e+31) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.sqrt(2.0) * 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 x_45_scale_m <= 1.35e+31:
		tmp = y_45_scale_m * b
	else:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.sqrt(2.0) * 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 (x_45_scale_m <= 1.35e+31)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * 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 (x_45_scale_m <= 1.35e+31)
		tmp = y_45_scale_m * b;
	else
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * 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[x$45$scale$95$m, 1.35e+31], N[(y$45$scale$95$m * b), $MachinePrecision], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.34999999999999993e31

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 19.2%

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

   5. Simplified 19.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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 19.3%

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

   7. Applied egg-rr 19.3%

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

   8. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 19.3%

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

   10. Simplified 19.3%

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

   if 1.34999999999999993e31 < x-scale

   1. Initial program 3.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 63.3%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

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

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

      3. sqrt-lowering-sqrt.f64 19.2%

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

   8. Simplified 19.2%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.35 \cdot 10^{+31}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 23.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 4.6 \cdot 10^{+28}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(a \cdot \left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \end{array} \]

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 4.6e+28)
   (* y-scale_m b)
   (* (sqrt 2.0) (* a (* 0.25 (* x-scale_m (sqrt 8.0)))))))

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 <= 4.6e+28) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = sqrt(2.0) * (a * (0.25 * (x_45_scale_m * sqrt(8.0))));
	}
	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 (x_45scale_m <= 4.6d+28) then
        tmp = y_45scale_m * b
    else
        tmp = sqrt(2.0d0) * (a * (0.25d0 * (x_45scale_m * sqrt(8.0d0))))
    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 (x_45_scale_m <= 4.6e+28) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = Math.sqrt(2.0) * (a * (0.25 * (x_45_scale_m * Math.sqrt(8.0))));
	}
	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 <= 4.6e+28:
		tmp = y_45_scale_m * b
	else:
		tmp = math.sqrt(2.0) * (a * (0.25 * (x_45_scale_m * math.sqrt(8.0))))
	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 <= 4.6e+28)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(sqrt(2.0) * Float64(a * Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0)))));
	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 <= 4.6e+28)
		tmp = y_45_scale_m * b;
	else
		tmp = sqrt(2.0) * (a * (0.25 * (x_45_scale_m * sqrt(8.0))));
	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, 4.6e+28], N[(y$45$scale$95$m * b), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(a * N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.59999999999999968e28

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 19.2%

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

   5. Simplified 19.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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 19.3%

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

   7. Applied egg-rr 19.3%

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

   8. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 19.3%

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

   10. Simplified 19.3%

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

   if 4.59999999999999968e28 < x-scale

   1. Initial program 3.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 63.3%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

   7. Applied egg-rr 68.1%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{*.f64}\left(\color{blue}{a}, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 19.2%

       \[\leadsto \sqrt{2} \cdot \left(\color{blue}{a} \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 4.6 \cdot 10^{+28}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(a \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 23.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 2.05 \cdot 10^{+37}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot \left(x-scale\_m \cdot a\right)\right)\\ \end{array} \end{array} \]

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 2.05e+37)
   (* y-scale_m b)
   (* 0.25 (* (* (sqrt 8.0) (sqrt 2.0)) (* 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 (x_45_scale_m <= 2.05e+37) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * ((sqrt(8.0) * sqrt(2.0)) * (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 (x_45scale_m <= 2.05d+37) then
        tmp = y_45scale_m * b
    else
        tmp = 0.25d0 * ((sqrt(8.0d0) * sqrt(2.0d0)) * (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 (x_45_scale_m <= 2.05e+37) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * ((Math.sqrt(8.0) * Math.sqrt(2.0)) * (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 x_45_scale_m <= 2.05e+37:
		tmp = y_45_scale_m * b
	else:
		tmp = 0.25 * ((math.sqrt(8.0) * math.sqrt(2.0)) * (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 (x_45_scale_m <= 2.05e+37)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * Float64(Float64(sqrt(8.0) * sqrt(2.0)) * 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 (x_45_scale_m <= 2.05e+37)
		tmp = y_45_scale_m * b;
	else
		tmp = 0.25 * ((sqrt(8.0) * sqrt(2.0)) * (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[x$45$scale$95$m, 2.05e+37], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.0499999999999999e37

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 19.2%

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

   5. Simplified 19.2%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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 19.3%

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

   7. Applied egg-rr 19.3%

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

   8. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 19.3%

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

   10. Simplified 19.3%

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

   if 2.0499999999999999e37 < x-scale

   1. Initial program 3.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 63.3%

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

   6. Taylor expanded in angle around 0

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

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

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 19.2%

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

   8. Simplified 19.2%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.05 \cdot 10^{+37}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot \left(x-scale \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 19.8% accurate, 23.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 5 \cdot 10^{+118}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale\_m \cdot 0.25\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\right)\\ \end{array} \end{array} \]

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 5e+118)
   (* y-scale_m b)
   (*
    (* x-scale_m 0.25)
    (* (sin (* 0.005555555555555556 (* angle PI))) (* b 4.0)))))

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 <= 5e+118) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (x_45_scale_m * 0.25) * (sin((0.005555555555555556 * (angle * ((double) M_PI)))) * (b * 4.0));
	}
	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 <= 5e+118) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (x_45_scale_m * 0.25) * (Math.sin((0.005555555555555556 * (angle * Math.PI))) * (b * 4.0));
	}
	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 <= 5e+118:
		tmp = y_45_scale_m * b
	else:
		tmp = (x_45_scale_m * 0.25) * (math.sin((0.005555555555555556 * (angle * math.pi))) * (b * 4.0))
	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 <= 5e+118)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(Float64(x_45_scale_m * 0.25) * Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(b * 4.0)));
	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 <= 5e+118)
		tmp = y_45_scale_m * b;
	else
		tmp = (x_45_scale_m * 0.25) * (sin((0.005555555555555556 * (angle * pi))) * (b * 4.0));
	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, 5e+118], N[(y$45$scale$95$m * b), $MachinePrecision], N[(N[(x$45$scale$95$m * 0.25), $MachinePrecision] * N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.99999999999999972e118

   1. Initial program 2.2%

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

   3. Taylor expanded in 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 20.4%

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

   5. Simplified 20.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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 20.6%

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

   7. Applied egg-rr 20.6%

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

   8. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 20.6%

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

   10. Simplified 20.6%

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

   if 4.99999999999999972e118 < x-scale

   1. Initial program 5.1%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 74.3%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. PI-lowering-PI.f64 32.4%

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

   8. Simplified 32.4%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot b\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. associate-*r* N/A

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

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

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

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

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

      8. sqrt-unprod N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(\sqrt{16}\right)\right), \sin \left(\frac{1}{180} \cdot \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      14. PI-lowering-PI.f64 27.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, 4\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 27.9%

       \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \left(\left(b \cdot 4\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 5 \cdot 10^{+118}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 19.8% accurate, 23.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 4.5 \cdot 10^{+117}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(x-scale\_m \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right)\right)\right)\\ \end{array} \end{array} \]

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 4.5e+117)
   (* y-scale_m b)
   (*
    0.25
    (* b (* x-scale_m (* (sin (* 0.005555555555555556 (* angle PI))) 4.0))))))

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 <= 4.5e+117) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (b * (x_45_scale_m * (sin((0.005555555555555556 * (angle * ((double) M_PI)))) * 4.0)));
	}
	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 <= 4.5e+117) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (b * (x_45_scale_m * (Math.sin((0.005555555555555556 * (angle * Math.PI))) * 4.0)));
	}
	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 <= 4.5e+117:
		tmp = y_45_scale_m * b
	else:
		tmp = 0.25 * (b * (x_45_scale_m * (math.sin((0.005555555555555556 * (angle * math.pi))) * 4.0)))
	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 <= 4.5e+117)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * Float64(b * Float64(x_45_scale_m * Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * 4.0))));
	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 <= 4.5e+117)
		tmp = y_45_scale_m * b;
	else
		tmp = 0.25 * (b * (x_45_scale_m * (sin((0.005555555555555556 * (angle * pi))) * 4.0)));
	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, 4.5e+117], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[(b * N[(x$45$scale$95$m * N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.5e117

   1. Initial program 2.2%

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

   3. Taylor expanded in 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 20.4%

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

   5. Simplified 20.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
   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 20.6%

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

   7. Applied egg-rr 20.6%

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

   8. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 20.6%

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

   10. Simplified 20.6%

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

   if 4.5e117 < x-scale

   1. Initial program 5.1%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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 74.3%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. PI-lowering-PI.f64 32.4%

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

   8. Simplified 32.4%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot b\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{8}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{16}\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), x-scale\right), b\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), x-scale\right), b\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), x-scale\right), b\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), x-scale\right), b\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right), x-scale\right), b\right)\right) \]

      12. PI-lowering-PI.f64 27.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), x-scale\right), b\right)\right) \]

   10. Applied egg-rr 27.9%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(4 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot x-scale\right) \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 4.5 \cdot 10^{+117}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 17.2% accurate, 919.0× speedup

Math

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

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 (* y-scale_m 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) {
	return y_45_scale_m * b;
}

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 = y_45scale_m * b
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 y_45_scale_m * b;
}

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 y_45_scale_m * b

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(y_45_scale_m * b)
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 = y_45_scale_m * b;
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[(y$45$scale$95$m * b), $MachinePrecision]

Derivation

1. Initial program 2.7%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
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 18.6%

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

5. Simplified 18.6%

   \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)} \]
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 18.6%

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

7. Applied egg-rr 18.6%

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

8. Taylor expanded in b around 0

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

   1. *-lowering-*.f64 18.6%

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

10. Simplified 18.6%

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

11. Final simplification 18.6%

    \[\leadsto y-scale \cdot b \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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))))