a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 55.3%

Time: 44.5s

Alternatives: 11

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: 55.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle}}\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ \mathbf{if}\;y-scale\_m \leq 7.4 \cdot 10^{+71}:\\ \;\;\;\;\left(\left(x-scale\_m \cdot \left(\sqrt{8} \cdot 0.25\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(b \cdot \sin t\_0, a \cdot \cos t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\left(b \cdot b\right) \cdot {\cos t\_1}^{2} + \left(a \cdot a\right) \cdot {\sin t\_1}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle)))
        (t_1 (* PI (* angle 0.005555555555555556))))
   (if (<= y-scale_m 7.4e+71)
     (*
      (* (* x-scale_m (* (sqrt 8.0) 0.25)) (sqrt 2.0))
      (hypot (* b (sin t_0)) (* a (cos t_0))))
     (*
      (* 0.25 (* y-scale_m (sqrt 8.0)))
      (sqrt
       (*
        2.0
        (+
         (* (* b b) (pow (cos t_1) 2.0))
         (* (* a a) (pow (sin t_1) 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 = ((double) M_PI) / (180.0 / angle);
	double t_1 = ((double) M_PI) * (angle * 0.005555555555555556);
	double tmp;
	if (y_45_scale_m <= 7.4e+71) {
		tmp = ((x_45_scale_m * (sqrt(8.0) * 0.25)) * sqrt(2.0)) * hypot((b * sin(t_0)), (a * cos(t_0)));
	} else {
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (((b * b) * pow(cos(t_1), 2.0)) + ((a * a) * pow(sin(t_1), 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 = Math.PI / (180.0 / angle);
	double t_1 = Math.PI * (angle * 0.005555555555555556);
	double tmp;
	if (y_45_scale_m <= 7.4e+71) {
		tmp = ((x_45_scale_m * (Math.sqrt(8.0) * 0.25)) * Math.sqrt(2.0)) * Math.hypot((b * Math.sin(t_0)), (a * Math.cos(t_0)));
	} else {
		tmp = (0.25 * (y_45_scale_m * Math.sqrt(8.0))) * Math.sqrt((2.0 * (((b * b) * Math.pow(Math.cos(t_1), 2.0)) + ((a * a) * Math.pow(Math.sin(t_1), 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 = math.pi / (180.0 / angle)
	t_1 = math.pi * (angle * 0.005555555555555556)
	tmp = 0
	if y_45_scale_m <= 7.4e+71:
		tmp = ((x_45_scale_m * (math.sqrt(8.0) * 0.25)) * math.sqrt(2.0)) * math.hypot((b * math.sin(t_0)), (a * math.cos(t_0)))
	else:
		tmp = (0.25 * (y_45_scale_m * math.sqrt(8.0))) * math.sqrt((2.0 * (((b * b) * math.pow(math.cos(t_1), 2.0)) + ((a * a) * math.pow(math.sin(t_1), 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(pi / Float64(180.0 / angle))
	t_1 = Float64(pi * Float64(angle * 0.005555555555555556))
	tmp = 0.0
	if (y_45_scale_m <= 7.4e+71)
		tmp = Float64(Float64(Float64(x_45_scale_m * Float64(sqrt(8.0) * 0.25)) * sqrt(2.0)) * hypot(Float64(b * sin(t_0)), Float64(a * cos(t_0))));
	else
		tmp = Float64(Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0))) * sqrt(Float64(2.0 * Float64(Float64(Float64(b * b) * (cos(t_1) ^ 2.0)) + Float64(Float64(a * a) * (sin(t_1) ^ 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 = pi / (180.0 / angle);
	t_1 = pi * (angle * 0.005555555555555556);
	tmp = 0.0;
	if (y_45_scale_m <= 7.4e+71)
		tmp = ((x_45_scale_m * (sqrt(8.0) * 0.25)) * sqrt(2.0)) * hypot((b * sin(t_0)), (a * cos(t_0)));
	else
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (((b * b) * (cos(t_1) ^ 2.0)) + ((a * a) * (sin(t_1) ^ 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[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 7.4e+71], N[(N[(N[(x$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $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], N[(N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(N[(b * b), $MachinePrecision] * N[Power[N[Cos[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.4e71

   1. Initial program 1.0%

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

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 25.3%

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

   9. Applied egg-rr 25.4%

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

       1. add-cube-cbrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       2. pow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       3. add-sqr-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       4. cbrt-prod N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       5. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       14. PI-lowering-PI.f64 25.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 25.4%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\left(\left(b \cdot b\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right) + \left(a \cdot a\right) \cdot {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\right)}\right)}^{2}\right)} \]

   13. Applied egg-rr 25.7%

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

   if 7.4e71 < y-scale

   1. Initial program 2.7%

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

   3. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 64.7%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 7.4 \cdot 10^{+71}:\\ \;\;\;\;\left(\left(x-scale \cdot \left(\sqrt{8} \cdot 0.25\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right), a \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\left(b \cdot b\right) \cdot {\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2} + \left(a \cdot a\right) \cdot {\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 42.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle}}\\ \mathbf{if}\;y-scale\_m \leq 1.7 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(x-scale\_m \cdot \left(\sqrt{8} \cdot 0.25\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(b \cdot \sin t\_0, a \cdot \cos t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(b \cdot \left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale\_m}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (/ PI (/ 180.0 angle))))
   (if (<= y-scale_m 1.7e+67)
     (*
      (* (* x-scale_m (* (sqrt 8.0) 0.25)) (sqrt 2.0))
      (hypot (* b (sin t_0)) (* a (cos t_0))))
     (*
      (* 0.25 (* b (* x-scale_m (* y-scale_m (sqrt 8.0)))))
      (/ (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 t_0 = ((double) M_PI) / (180.0 / angle);
	double tmp;
	if (y_45_scale_m <= 1.7e+67) {
		tmp = ((x_45_scale_m * (sqrt(8.0) * 0.25)) * sqrt(2.0)) * hypot((b * sin(t_0)), (a * cos(t_0)));
	} else {
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * sqrt(8.0))))) * (sqrt(2.0) / x_45_scale_m);
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.PI / (180.0 / angle);
	double tmp;
	if (y_45_scale_m <= 1.7e+67) {
		tmp = ((x_45_scale_m * (Math.sqrt(8.0) * 0.25)) * Math.sqrt(2.0)) * Math.hypot((b * Math.sin(t_0)), (a * Math.cos(t_0)));
	} else {
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * Math.sqrt(8.0))))) * (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):
	t_0 = math.pi / (180.0 / angle)
	tmp = 0
	if y_45_scale_m <= 1.7e+67:
		tmp = ((x_45_scale_m * (math.sqrt(8.0) * 0.25)) * math.sqrt(2.0)) * math.hypot((b * math.sin(t_0)), (a * math.cos(t_0)))
	else:
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * math.sqrt(8.0))))) * (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)
	t_0 = Float64(pi / Float64(180.0 / angle))
	tmp = 0.0
	if (y_45_scale_m <= 1.7e+67)
		tmp = Float64(Float64(Float64(x_45_scale_m * Float64(sqrt(8.0) * 0.25)) * sqrt(2.0)) * hypot(Float64(b * sin(t_0)), Float64(a * cos(t_0))));
	else
		tmp = Float64(Float64(0.25 * Float64(b * Float64(x_45_scale_m * Float64(y_45_scale_m * sqrt(8.0))))) * 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)
	t_0 = pi / (180.0 / angle);
	tmp = 0.0;
	if (y_45_scale_m <= 1.7e+67)
		tmp = ((x_45_scale_m * (sqrt(8.0) * 0.25)) * sqrt(2.0)) * hypot((b * sin(t_0)), (a * cos(t_0)));
	else
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * sqrt(8.0))))) * (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_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 1.7e+67], N[(N[(N[(x$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $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], N[(N[(0.25 * N[(b * N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $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 y-scale < 1.7000000000000001e67

   1. Initial program 1.0%

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

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 25.4%

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

   9. Applied egg-rr 25.5%

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

       1. add-cube-cbrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       2. pow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       3. add-sqr-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       4. cbrt-prod N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       5. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       14. PI-lowering-PI.f64 25.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 25.5%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\left(\left(b \cdot b\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right) + \left(a \cdot a\right) \cdot {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\right)}\right)}^{2}\right)} \]

   13. Applied egg-rr 25.9%

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

   if 1.7000000000000001e67 < y-scale

   1. Initial program 2.7%

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

   3. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

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

   7. Simplified 9.6%

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

   8. Taylor expanded in angle around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 30.9%

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

   10. Simplified 30.9%

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

Alternative 3: 42.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{\pi \cdot angle}{180}\\ \mathbf{if}\;y-scale\_m \leq 2.1 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{hypot}\left(a \cdot \cos t\_0, b \cdot \sin t\_0\right) \cdot \left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(b \cdot \left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale\_m}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (/ (* PI angle) 180.0)))
   (if (<= y-scale_m 2.1e+65)
     (*
      (sqrt 2.0)
      (*
       (hypot (* a (cos t_0)) (* b (sin t_0)))
       (* 0.25 (* x-scale_m (sqrt 8.0)))))
     (*
      (* 0.25 (* b (* x-scale_m (* y-scale_m (sqrt 8.0)))))
      (/ (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 t_0 = (((double) M_PI) * angle) / 180.0;
	double tmp;
	if (y_45_scale_m <= 2.1e+65) {
		tmp = sqrt(2.0) * (hypot((a * cos(t_0)), (b * sin(t_0))) * (0.25 * (x_45_scale_m * sqrt(8.0))));
	} else {
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * sqrt(8.0))))) * (sqrt(2.0) / x_45_scale_m);
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (Math.PI * angle) / 180.0;
	double tmp;
	if (y_45_scale_m <= 2.1e+65) {
		tmp = Math.sqrt(2.0) * (Math.hypot((a * Math.cos(t_0)), (b * Math.sin(t_0))) * (0.25 * (x_45_scale_m * Math.sqrt(8.0))));
	} else {
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * Math.sqrt(8.0))))) * (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):
	t_0 = (math.pi * angle) / 180.0
	tmp = 0
	if y_45_scale_m <= 2.1e+65:
		tmp = math.sqrt(2.0) * (math.hypot((a * math.cos(t_0)), (b * math.sin(t_0))) * (0.25 * (x_45_scale_m * math.sqrt(8.0))))
	else:
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * math.sqrt(8.0))))) * (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)
	t_0 = Float64(Float64(pi * angle) / 180.0)
	tmp = 0.0
	if (y_45_scale_m <= 2.1e+65)
		tmp = Float64(sqrt(2.0) * Float64(hypot(Float64(a * cos(t_0)), Float64(b * sin(t_0))) * Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0)))));
	else
		tmp = Float64(Float64(0.25 * Float64(b * Float64(x_45_scale_m * Float64(y_45_scale_m * sqrt(8.0))))) * 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)
	t_0 = (pi * angle) / 180.0;
	tmp = 0.0;
	if (y_45_scale_m <= 2.1e+65)
		tmp = sqrt(2.0) * (hypot((a * cos(t_0)), (b * sin(t_0))) * (0.25 * (x_45_scale_m * sqrt(8.0))));
	else
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * sqrt(8.0))))) * (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_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] / 180.0), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.1e+65], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(b * N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $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 y-scale < 2.09999999999999991e65

   1. Initial program 1.0%

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

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 25.4%

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

   9. Applied egg-rr 25.5%

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

   11. Applied egg-rr 25.9%

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

   if 2.09999999999999991e65 < y-scale

   1. Initial program 2.7%

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

   3. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

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

   7. Simplified 9.6%

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

   8. Taylor expanded in angle around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 30.9%

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

   10. Simplified 30.9%

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

Alternative 4: 36.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 300000000000:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) + {\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}\\ \end{array} \end{array} \]

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 300000000000.0)
   (* y-scale_m b)
   (*
    (* 0.25 (* x-scale_m (sqrt 8.0)))
    (sqrt
     (*
      2.0
      (+
       (* (* 3.08641975308642e-5 (* angle angle)) (* (* b b) (* PI PI)))
       (* (pow (cos (* PI (* angle 0.005555555555555556))) 2.0) (* a 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 <= 300000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (((double) M_PI) * ((double) M_PI)))) + (pow(cos((((double) M_PI) * (angle * 0.005555555555555556))), 2.0) * (a * 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 <= 300000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.sqrt((2.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (Math.PI * Math.PI))) + (Math.pow(Math.cos((Math.PI * (angle * 0.005555555555555556))), 2.0) * (a * 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 <= 300000000000.0:
		tmp = y_45_scale_m * b
	else:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.sqrt((2.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (math.pi * math.pi))) + (math.pow(math.cos((math.pi * (angle * 0.005555555555555556))), 2.0) * (a * 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 <= 300000000000.0)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * sqrt(Float64(2.0 * Float64(Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(Float64(b * b) * Float64(pi * pi))) + Float64((cos(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 2.0) * Float64(a * 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 <= 300000000000.0)
		tmp = y_45_scale_m * b;
	else
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (pi * pi))) + ((cos((pi * (angle * 0.005555555555555556))) ^ 2.0) * (a * 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, 300000000000.0], 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[Sqrt[N[(2.0 * N[(N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3e11

   1. Initial program 1.6%

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

   3. Simplified 1.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. 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 22.1%

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

   7. Simplified 22.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 22.3%

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

   9. Applied egg-rr 22.3%

      \[\leadsto \color{blue}{y-scale \cdot \left(4 \cdot \left(0.25 \cdot b\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. *-lft-identity 22.3%

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

   11. Applied egg-rr 22.3%

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

   if 3e11 < x-scale

   1. Initial program 0.3%

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

   3. Simplified 0.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 67.1%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \left({angle}^{2}\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot angle\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      12. PI-lowering-PI.f64 65.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   10. Simplified 65.3%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right)} + \left(a \cdot a\right) \cdot {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 300000000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right) + {\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{if}\;b \leq 9.2 \cdot 10^{+48}:\\ \;\;\;\;0.25 \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left(2 \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot t\_0\right) + b \cdot \left(b \cdot \left(0.5 + t\_0 \cdot -0.5\right)\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{\sqrt{2}}{x-scale\_m} \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(x-scale\_m \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

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 (cos (* (* PI angle) 0.011111111111111112))))
   (if (<= b 9.2e+48)
     (*
      0.25
      (*
       x-scale_m
       (sqrt
        (*
         8.0
         (*
          2.0
          (+
           (* (* a a) (+ 0.5 (* 0.5 t_0)))
           (* b (* b (+ 0.5 (* t_0 -0.5))))))))))
     (*
      0.25
      (*
       (/ (sqrt 2.0) x-scale_m)
       (* (* y-scale_m (sqrt 8.0)) (* x-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) {
	double t_0 = cos(((((double) M_PI) * angle) * 0.011111111111111112));
	double tmp;
	if (b <= 9.2e+48) {
		tmp = 0.25 * (x_45_scale_m * sqrt((8.0 * (2.0 * (((a * a) * (0.5 + (0.5 * t_0))) + (b * (b * (0.5 + (t_0 * -0.5)))))))));
	} else {
		tmp = 0.25 * ((sqrt(2.0) / x_45_scale_m) * ((y_45_scale_m * sqrt(8.0)) * (x_45_scale_m * b)));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.cos(((Math.PI * angle) * 0.011111111111111112));
	double tmp;
	if (b <= 9.2e+48) {
		tmp = 0.25 * (x_45_scale_m * Math.sqrt((8.0 * (2.0 * (((a * a) * (0.5 + (0.5 * t_0))) + (b * (b * (0.5 + (t_0 * -0.5)))))))));
	} else {
		tmp = 0.25 * ((Math.sqrt(2.0) / x_45_scale_m) * ((y_45_scale_m * Math.sqrt(8.0)) * (x_45_scale_m * b)));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.cos(((math.pi * angle) * 0.011111111111111112))
	tmp = 0
	if b <= 9.2e+48:
		tmp = 0.25 * (x_45_scale_m * math.sqrt((8.0 * (2.0 * (((a * a) * (0.5 + (0.5 * t_0))) + (b * (b * (0.5 + (t_0 * -0.5)))))))))
	else:
		tmp = 0.25 * ((math.sqrt(2.0) / x_45_scale_m) * ((y_45_scale_m * math.sqrt(8.0)) * (x_45_scale_m * b)))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = cos(Float64(Float64(pi * angle) * 0.011111111111111112))
	tmp = 0.0
	if (b <= 9.2e+48)
		tmp = Float64(0.25 * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64(2.0 * Float64(Float64(Float64(a * a) * Float64(0.5 + Float64(0.5 * t_0))) + Float64(b * Float64(b * Float64(0.5 + Float64(t_0 * -0.5))))))))));
	else
		tmp = Float64(0.25 * Float64(Float64(sqrt(2.0) / x_45_scale_m) * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(x_45_scale_m * b))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = cos(((pi * angle) * 0.011111111111111112));
	tmp = 0.0;
	if (b <= 9.2e+48)
		tmp = 0.25 * (x_45_scale_m * sqrt((8.0 * (2.0 * (((a * a) * (0.5 + (0.5 * t_0))) + (b * (b * (0.5 + (t_0 * -0.5)))))))));
	else
		tmp = 0.25 * ((sqrt(2.0) / x_45_scale_m) * ((y_45_scale_m * sqrt(8.0)) * (x_45_scale_m * b)));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 9.2e+48], N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(0.5 + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(0.5 + N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 9.2000000000000001e48

   1. Initial program 1.4%

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

   3. Simplified 1.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 25.5%

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. div-inv N/A

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

      10. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

   9. Applied egg-rr 25.9%

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

       1. add-cube-cbrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       2. pow3 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       3. add-sqr-sqrt N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       4. cbrt-prod N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       5. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\left(\sqrt{\mathsf{PI}\left(\right)}\right)\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       14. PI-lowering-PI.f64 25.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, angle\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right), \mathsf{pow.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), 3\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 25.9%

       \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(\left(\left(b \cdot b\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right) + \left(a \cdot a\right) \cdot {\cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\right)}\right)}^{2}\right)} \]

   13. Applied egg-rr 23.5%

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

   if 9.2000000000000001e48 < b

   1. Initial program 0.8%

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

   3. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

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

   7. Simplified 18.6%

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

   9. Applied egg-rr 19.0%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{8} \cdot y-scale\right) \cdot \left(x-scale \cdot b\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{0.5 + 0.5 \cdot \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right)}{\frac{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}{0.5 + \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right) \cdot -0.5}} + {\left(\frac{0.5 + 0.5 \cdot \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right)}{x-scale \cdot x-scale} - \frac{0.5 + \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right) \cdot -0.5}{y-scale \cdot y-scale}\right)}^{2}} + \left(\frac{0.5 + 0.5 \cdot \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right)}{x-scale \cdot x-scale} + \frac{0.5 + \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right) \cdot -0.5}{y-scale \cdot y-scale}\right)}\right) \cdot 0.25} \]

   10. Taylor expanded in angle around 0

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

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

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

       2. sqrt-lowering-sqrt.f64 44.3%

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

   12. Simplified 44.3%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{+48}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \sqrt{8 \cdot \left(2 \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) + b \cdot \left(b \cdot \left(0.5 + \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot -0.5\right)\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{\sqrt{2}}{x-scale} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 28.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{\sqrt{2}}{x-scale\_m} \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(x-scale\_m \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= b 3.8e-7)
   (* (sqrt 2.0) (* 0.25 (* a (* x-scale_m (sqrt 8.0)))))
   (*
    0.25
    (*
     (/ (sqrt 2.0) x-scale_m)
     (* (* y-scale_m (sqrt 8.0)) (* x-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) {
	double tmp;
	if (b <= 3.8e-7) {
		tmp = sqrt(2.0) * (0.25 * (a * (x_45_scale_m * sqrt(8.0))));
	} else {
		tmp = 0.25 * ((sqrt(2.0) / x_45_scale_m) * ((y_45_scale_m * sqrt(8.0)) * (x_45_scale_m * b)));
	}
	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 <= 3.8d-7) then
        tmp = sqrt(2.0d0) * (0.25d0 * (a * (x_45scale_m * sqrt(8.0d0))))
    else
        tmp = 0.25d0 * ((sqrt(2.0d0) / x_45scale_m) * ((y_45scale_m * sqrt(8.0d0)) * (x_45scale_m * b)))
    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 <= 3.8e-7) {
		tmp = Math.sqrt(2.0) * (0.25 * (a * (x_45_scale_m * Math.sqrt(8.0))));
	} else {
		tmp = 0.25 * ((Math.sqrt(2.0) / x_45_scale_m) * ((y_45_scale_m * Math.sqrt(8.0)) * (x_45_scale_m * b)));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 3.8e-7:
		tmp = math.sqrt(2.0) * (0.25 * (a * (x_45_scale_m * math.sqrt(8.0))))
	else:
		tmp = 0.25 * ((math.sqrt(2.0) / x_45_scale_m) * ((y_45_scale_m * math.sqrt(8.0)) * (x_45_scale_m * b)))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 3.8e-7)
		tmp = Float64(sqrt(2.0) * Float64(0.25 * Float64(a * Float64(x_45_scale_m * sqrt(8.0)))));
	else
		tmp = Float64(0.25 * Float64(Float64(sqrt(2.0) / x_45_scale_m) * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(x_45_scale_m * b))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 3.8e-7)
		tmp = sqrt(2.0) * (0.25 * (a * (x_45_scale_m * sqrt(8.0))));
	else
		tmp = 0.25 * ((sqrt(2.0) / x_45_scale_m) * ((y_45_scale_m * sqrt(8.0)) * (x_45_scale_m * b)));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 3.8e-7], N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 * N[(a * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.80000000000000015e-7

   1. Initial program 1.5%

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

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 27.0%

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

   8. Taylor expanded in b around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 19.1%

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

   10. Simplified 19.1%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       13. associate-*r* N/A

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

       14. metadata-eval N/A

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

       15. associate-/r/ N/A

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

   12. Applied egg-rr 18.1%

       \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right) \cdot \sqrt{2}\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)} \]

   13. Taylor expanded in angle around 0

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

   15. Simplified 17.9%

       \[\leadsto \left(\left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right) \cdot \sqrt{2}\right) \cdot \color{blue}{1} \]

   if 3.80000000000000015e-7 < b

   1. Initial program 0.8%

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

   3. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

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

   7. Simplified 15.0%

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

   9. Applied egg-rr 17.2%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{8} \cdot y-scale\right) \cdot \left(x-scale \cdot b\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{0.5 + 0.5 \cdot \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right)}{\frac{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}{0.5 + \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right) \cdot -0.5}} + {\left(\frac{0.5 + 0.5 \cdot \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right)}{x-scale \cdot x-scale} - \frac{0.5 + \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right) \cdot -0.5}{y-scale \cdot y-scale}\right)}^{2}} + \left(\frac{0.5 + 0.5 \cdot \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right)}{x-scale \cdot x-scale} + \frac{0.5 + \cos \left(\frac{2}{\frac{180}{angle \cdot \pi}}\right) \cdot -0.5}{y-scale \cdot y-scale}\right)}\right) \cdot 0.25} \]

   10. Taylor expanded in angle around 0

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

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

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

       2. sqrt-lowering-sqrt.f64 39.9%

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

   12. Simplified 39.9%

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

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{\sqrt{2}}{x-scale} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 24.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 3.7 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(b \cdot \left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale\_m}\\ \end{array} \end{array} \]

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 (<= y-scale_m 3.7e+68)
   (* (sqrt 2.0) (* 0.25 (* a (* x-scale_m (sqrt 8.0)))))
   (*
    (* 0.25 (* b (* x-scale_m (* y-scale_m (sqrt 8.0)))))
    (/ (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 (y_45_scale_m <= 3.7e+68) {
		tmp = sqrt(2.0) * (0.25 * (a * (x_45_scale_m * sqrt(8.0))));
	} else {
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * sqrt(8.0))))) * (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 (y_45scale_m <= 3.7d+68) then
        tmp = sqrt(2.0d0) * (0.25d0 * (a * (x_45scale_m * sqrt(8.0d0))))
    else
        tmp = (0.25d0 * (b * (x_45scale_m * (y_45scale_m * sqrt(8.0d0))))) * (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 (y_45_scale_m <= 3.7e+68) {
		tmp = Math.sqrt(2.0) * (0.25 * (a * (x_45_scale_m * Math.sqrt(8.0))));
	} else {
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * Math.sqrt(8.0))))) * (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 y_45_scale_m <= 3.7e+68:
		tmp = math.sqrt(2.0) * (0.25 * (a * (x_45_scale_m * math.sqrt(8.0))))
	else:
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * math.sqrt(8.0))))) * (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 (y_45_scale_m <= 3.7e+68)
		tmp = Float64(sqrt(2.0) * Float64(0.25 * Float64(a * Float64(x_45_scale_m * sqrt(8.0)))));
	else
		tmp = Float64(Float64(0.25 * Float64(b * Float64(x_45_scale_m * Float64(y_45_scale_m * sqrt(8.0))))) * 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 (y_45_scale_m <= 3.7e+68)
		tmp = sqrt(2.0) * (0.25 * (a * (x_45_scale_m * sqrt(8.0))));
	else
		tmp = (0.25 * (b * (x_45_scale_m * (y_45_scale_m * sqrt(8.0))))) * (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[y$45$scale$95$m, 3.7e+68], N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 * N[(a * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(b * N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $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 y-scale < 3.69999999999999998e68

   1. Initial program 1.0%

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

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 25.4%

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

   8. Taylor expanded in b around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 19.7%

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

   10. Simplified 19.7%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       13. associate-*r* N/A

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

       14. metadata-eval N/A

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

       15. associate-/r/ N/A

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

   12. Applied egg-rr 18.6%

       \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right) \cdot \sqrt{2}\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)} \]

   13. Taylor expanded in angle around 0

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

   15. Simplified 18.0%

       \[\leadsto \left(\left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right) \cdot \sqrt{2}\right) \cdot \color{blue}{1} \]

   if 3.69999999999999998e68 < y-scale

   1. Initial program 2.7%

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

   3. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf

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

   7. Simplified 9.6%

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

   8. Taylor expanded in angle around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 30.9%

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

   10. Simplified 30.9%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.7 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(b \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 25.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

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 1.12e+24)
   (* (sqrt 2.0) (* 0.25 (* a (* x-scale_m (sqrt 8.0)))))
   (* 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) {
	double tmp;
	if (b <= 1.12e+24) {
		tmp = sqrt(2.0) * (0.25 * (a * (x_45_scale_m * sqrt(8.0))));
	} else {
		tmp = y_45_scale_m * b;
	}
	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 <= 1.12d+24) then
        tmp = sqrt(2.0d0) * (0.25d0 * (a * (x_45scale_m * sqrt(8.0d0))))
    else
        tmp = y_45scale_m * b
    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 <= 1.12e+24) {
		tmp = Math.sqrt(2.0) * (0.25 * (a * (x_45_scale_m * Math.sqrt(8.0))));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 1.12e+24:
		tmp = math.sqrt(2.0) * (0.25 * (a * (x_45_scale_m * math.sqrt(8.0))))
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 1.12e+24)
		tmp = Float64(sqrt(2.0) * Float64(0.25 * Float64(a * Float64(x_45_scale_m * sqrt(8.0)))));
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 1.12e+24)
		tmp = sqrt(2.0) * (0.25 * (a * (x_45_scale_m * sqrt(8.0))));
	else
		tmp = y_45_scale_m * b;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 1.12e+24], N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 * N[(a * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.12e24

   1. Initial program 1.4%

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

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 26.0%

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

   8. Taylor expanded in b around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 19.0%

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

   10. Simplified 19.0%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       13. associate-*r* N/A

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

       14. metadata-eval N/A

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

       15. associate-/r/ N/A

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

   12. Applied egg-rr 18.0%

       \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right) \cdot \sqrt{2}\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)} \]

   13. Taylor expanded in angle around 0

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

   15. Simplified 17.8%

       \[\leadsto \left(\left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot a\right)\right) \cdot \sqrt{2}\right) \cdot \color{blue}{1} \]

   if 1.12e24 < b

   1. Initial program 1.0%

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

   3. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. 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 30.4%

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

   7. Simplified 30.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 30.5%

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

   9. Applied egg-rr 30.5%

      \[\leadsto \color{blue}{y-scale \cdot \left(4 \cdot \left(0.25 \cdot b\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. *-lft-identity 30.5%

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

   11. Applied egg-rr 30.5%

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

4. Final simplification 20.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.12 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2} \cdot \left(0.25 \cdot \left(a \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 25.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

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 4.6e+23)
   (* (* 0.25 (* x-scale_m (sqrt 8.0))) (* (sqrt 2.0) a))
   (* 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) {
	double tmp;
	if (b <= 4.6e+23) {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	} else {
		tmp = y_45_scale_m * b;
	}
	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 <= 4.6d+23) then
        tmp = (0.25d0 * (x_45scale_m * sqrt(8.0d0))) * (sqrt(2.0d0) * a)
    else
        tmp = y_45scale_m * b
    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 <= 4.6e+23) {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.sqrt(2.0) * a);
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 4.6e+23:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.sqrt(2.0) * a)
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 4.6e+23)
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * a));
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 4.6e+23)
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	else
		tmp = y_45_scale_m * b;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 4.6e+23], 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[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.6000000000000001e23

   1. Initial program 1.4%

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

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 26.0%

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

   8. Taylor expanded in angle around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 17.8%

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

   10. Simplified 17.8%

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

   if 4.6000000000000001e23 < b

   1. Initial program 1.0%

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

   3. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. 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 30.4%

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

   7. Simplified 30.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 30.5%

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

   9. Applied egg-rr 30.5%

      \[\leadsto \color{blue}{y-scale \cdot \left(4 \cdot \left(0.25 \cdot b\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. *-lft-identity 30.5%

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

   11. Applied egg-rr 30.5%

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

4. Final simplification 20.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 25.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+22}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot a\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

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 1.25e+22)
   (* 0.25 (* (* x-scale_m a) (* (sqrt 8.0) (sqrt 2.0))))
   (* 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) {
	double tmp;
	if (b <= 1.25e+22) {
		tmp = 0.25 * ((x_45_scale_m * a) * (sqrt(8.0) * sqrt(2.0)));
	} else {
		tmp = y_45_scale_m * b;
	}
	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 <= 1.25d+22) then
        tmp = 0.25d0 * ((x_45scale_m * a) * (sqrt(8.0d0) * sqrt(2.0d0)))
    else
        tmp = y_45scale_m * b
    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 <= 1.25e+22) {
		tmp = 0.25 * ((x_45_scale_m * a) * (Math.sqrt(8.0) * Math.sqrt(2.0)));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 1.25e+22:
		tmp = 0.25 * ((x_45_scale_m * a) * (math.sqrt(8.0) * math.sqrt(2.0)))
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 1.25e+22)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * a) * Float64(sqrt(8.0) * sqrt(2.0))));
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 1.25e+22)
		tmp = 0.25 * ((x_45_scale_m * a) * (sqrt(8.0) * sqrt(2.0)));
	else
		tmp = y_45_scale_m * b;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 1.25e+22], N[(0.25 * N[(N[(x$45$scale$95$m * a), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2499999999999999e22

   1. Initial program 1.4%

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

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)} \]
   6. 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

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

   7. Simplified 26.0%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   9. 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 17.7%

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

   10. Simplified 17.7%

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

   if 1.2499999999999999e22 < b

   1. Initial program 1.0%

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

   3. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. 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 30.4%

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

   7. Simplified 30.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 30.5%

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

   9. Applied egg-rr 30.5%

      \[\leadsto \color{blue}{y-scale \cdot \left(4 \cdot \left(0.25 \cdot b\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. metadata-eval N/A

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

       3. *-lft-identity 30.5%

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

   11. Applied egg-rr 30.5%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+22}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot a\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 17.0% 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 1.3%

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

3. Simplified 1.6%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{8 \cdot \left(\left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(\frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale} + \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{y-scale \cdot y-scale}, \sin \left(\frac{angle \cdot \pi}{180}\right) \cdot \left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \frac{\cos \left(\frac{angle \cdot \pi}{180}\right)}{x-scale \cdot y-scale}\right)\right)\right)\right)}}{\frac{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
6. 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 19.6%

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

7. Simplified 19.6%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. sqrt-unprod N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. *-lowering-*.f64 19.7%

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

9. Applied egg-rr 19.7%

   \[\leadsto \color{blue}{y-scale \cdot \left(4 \cdot \left(0.25 \cdot b\right)\right)} \]
10. Step-by-step derivation

    1. associate-*r* N/A

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

    2. metadata-eval N/A

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

    3. *-lft-identity 19.7%

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

11. Applied egg-rr 19.7%

    \[\leadsto y-scale \cdot \color{blue}{b} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(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))))