a from scale-rotated-ellipse

Percentage Accurate: 2.7% → 59.5%

Time: 38.6s

Alternatives: 13

Speedup: 919.0×

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

Specification

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (+
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) + sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) + sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 2.7% 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: 59.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{angle \cdot \pi}{180}\\ t_1 := \sin t\_0\\ \mathbf{if}\;y-scale\_m \leq 0.2:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{hypot}\left(a \cdot \cos t\_0, b \cdot t\_1\right) \cdot \left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot t\_1, b\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ (* angle PI) 180.0)) (t_1 (sin t_0)))
   (if (<= y-scale_m 0.2)
     (*
      (sqrt 2.0)
      (* (hypot (* a (cos t_0)) (* b t_1)) (* 0.25 (* x-scale_m (sqrt 8.0)))))
     (* 0.25 (* (* y-scale_m 4.0) (hypot (* a t_1) 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 = (angle * ((double) M_PI)) / 180.0;
	double t_1 = sin(t_0);
	double tmp;
	if (y_45_scale_m <= 0.2) {
		tmp = sqrt(2.0) * (hypot((a * cos(t_0)), (b * t_1)) * (0.25 * (x_45_scale_m * sqrt(8.0))));
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * hypot((a * t_1), 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 = (angle * Math.PI) / 180.0;
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y_45_scale_m <= 0.2) {
		tmp = Math.sqrt(2.0) * (Math.hypot((a * Math.cos(t_0)), (b * t_1)) * (0.25 * (x_45_scale_m * Math.sqrt(8.0))));
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * Math.hypot((a * t_1), 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 = (angle * math.pi) / 180.0
	t_1 = math.sin(t_0)
	tmp = 0
	if y_45_scale_m <= 0.2:
		tmp = math.sqrt(2.0) * (math.hypot((a * math.cos(t_0)), (b * t_1)) * (0.25 * (x_45_scale_m * math.sqrt(8.0))))
	else:
		tmp = 0.25 * ((y_45_scale_m * 4.0) * math.hypot((a * t_1), b))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(angle * pi) / 180.0)
	t_1 = sin(t_0)
	tmp = 0.0
	if (y_45_scale_m <= 0.2)
		tmp = Float64(sqrt(2.0) * Float64(hypot(Float64(a * cos(t_0)), Float64(b * t_1)) * Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0)))));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * 4.0) * hypot(Float64(a * t_1), 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 = (angle * pi) / 180.0;
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y_45_scale_m <= 0.2)
		tmp = sqrt(2.0) * (hypot((a * cos(t_0)), (b * t_1)) * (0.25 * (x_45_scale_m * sqrt(8.0))));
	else
		tmp = 0.25 * ((y_45_scale_m * 4.0) * hypot((a * t_1), 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[(N[(angle * Pi), $MachinePrecision] / 180.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 0.2], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * t$95$1), $MachinePrecision] ^ 2], $MachinePrecision] * N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 0.20000000000000001

   1. Initial program 3.1%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 24.6%

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

   7. Applied egg-rr 28.6%

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

   if 0.20000000000000001 < y-scale

   1. Initial program 1.7%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 53.8%

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

   7. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right)\right)\right), \frac{1}{4}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      18. PI-lowering-PI.f64 58.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 58.7%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)}\right)\right) \cdot 0.25 \]

   11. Taylor expanded in angle around 0

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

   13. Simplified 63.4%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), \color{blue}{b}\right)\right) \cdot 0.25 \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 0.2:\\ \;\;\;\;\sqrt{2} \cdot \left(\mathsf{hypot}\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right), b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right) \cdot \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(a \cdot a\right) + \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 1.05e+54)
   (* 0.25 (* (* y-scale_m 4.0) (hypot (* a (sin (/ (* angle PI) 180.0))) b)))
   (*
    (* 0.25 (* x-scale_m (sqrt 8.0)))
    (sqrt
     (*
      2.0
      (+
       (* (pow (cos (* (* angle PI) 0.005555555555555556)) 2.0) (* a a))
       (* (* 3.08641975308642e-5 (* angle angle)) (* (* b b) (* PI PI)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 1.05e+54) {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * hypot((a * sin(((angle * ((double) M_PI)) / 180.0))), b));
	} else {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * ((pow(cos(((angle * ((double) M_PI)) * 0.005555555555555556)), 2.0) * (a * a)) + ((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (((double) M_PI) * ((double) M_PI)))))));
	}
	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 <= 1.05e+54) {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * Math.hypot((a * Math.sin(((angle * Math.PI) / 180.0))), b));
	} else {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.sqrt((2.0 * ((Math.pow(Math.cos(((angle * Math.PI) * 0.005555555555555556)), 2.0) * (a * a)) + ((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (Math.PI * Math.PI))))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 1.05e+54:
		tmp = 0.25 * ((y_45_scale_m * 4.0) * math.hypot((a * math.sin(((angle * math.pi) / 180.0))), b))
	else:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.sqrt((2.0 * ((math.pow(math.cos(((angle * math.pi) * 0.005555555555555556)), 2.0) * (a * a)) + ((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (math.pi * math.pi))))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 1.05e+54)
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * 4.0) * hypot(Float64(a * sin(Float64(Float64(angle * pi) / 180.0))), b)));
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * sqrt(Float64(2.0 * Float64(Float64((cos(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 2.0) * Float64(a * a)) + Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(Float64(b * b) * Float64(pi * pi)))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 1.05e+54)
		tmp = 0.25 * ((y_45_scale_m * 4.0) * hypot((a * sin(((angle * pi) / 180.0))), b));
	else
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (((cos(((angle * pi) * 0.005555555555555556)) ^ 2.0) * (a * a)) + ((3.08641975308642e-5 * (angle * angle)) * ((b * b) * (pi * pi))))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 1.05e+54], N[(0.25 * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(angle * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(N[Power[N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.04999999999999993e54

   1. Initial program 2.4%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 22.6%

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

   7. Applied egg-rr 28.6%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right)\right)\right), \frac{1}{4}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      18. PI-lowering-PI.f64 27.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 27.7%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)}\right)\right) \cdot 0.25 \]

   11. Taylor expanded in angle around 0

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

   13. Simplified 29.1%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), \color{blue}{b}\right)\right) \cdot 0.25 \]

   if 1.04999999999999993e54 < x-scale

   1. Initial program 3.7%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 63.9%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. PI-lowering-PI.f64 59.1%

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

   8. Simplified 59.1%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\cos \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{2} \cdot \left(a \cdot a\right) + \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\pi \cdot \pi\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.8% accurate, 12.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 0.00155:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 0.00155)
   (* (* 0.25 (* x-scale_m (sqrt 8.0))) (* (sqrt 2.0) a))
   (*
    0.25
    (* (* y-scale_m 4.0) (hypot (* a (sin (/ (* angle PI) 180.0))) 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 (y_45_scale_m <= 0.00155) {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * hypot((a * sin(((angle * ((double) M_PI)) / 180.0))), 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 tmp;
	if (y_45_scale_m <= 0.00155) {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.sqrt(2.0) * a);
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * Math.hypot((a * Math.sin(((angle * Math.PI) / 180.0))), 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 y_45_scale_m <= 0.00155:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.sqrt(2.0) * a)
	else:
		tmp = 0.25 * ((y_45_scale_m * 4.0) * math.hypot((a * math.sin(((angle * math.pi) / 180.0))), 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 (y_45_scale_m <= 0.00155)
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * a));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * 4.0) * hypot(Float64(a * sin(Float64(Float64(angle * pi) / 180.0))), 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 (y_45_scale_m <= 0.00155)
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	else
		tmp = 0.25 * ((y_45_scale_m * 4.0) * hypot((a * sin(((angle * pi) / 180.0))), 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[y$45$scale$95$m, 0.00155], 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[(0.25 * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(angle * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 0.00154999999999999995

   1. Initial program 3.1%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 24.2%

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

   6. Taylor expanded in angle around 0

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

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

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

      2. sqrt-lowering-sqrt.f64 20.7%

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

   8. Simplified 20.7%

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

   if 0.00154999999999999995 < y-scale

   1. Initial program 1.7%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 53.1%

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

   7. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right)\right)\right), \frac{1}{4}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      18. PI-lowering-PI.f64 57.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 57.9%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)}\right)\right) \cdot 0.25 \]

   11. Taylor expanded in angle around 0

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

   13. Simplified 62.5%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), \color{blue}{b}\right)\right) \cdot 0.25 \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 0.00155:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 25.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.75 \cdot 10^{+16}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \left(b \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 1.75e+16)
   (* (* 0.25 (* x-scale_m (sqrt 8.0))) (* (sqrt 2.0) a))
   (*
    0.25
    (*
     (* y-scale_m 4.0)
     (*
      b
      (+
       1.0
       (*
        (* angle angle)
        (+
         (* (* PI PI) -1.54320987654321e-5)
         (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0))))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.75e+16) {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0)))))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.75e+16) {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.sqrt(2.0) * a);
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0)))))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 1.75e+16:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.sqrt(2.0) * a)
	else:
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0)))))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 1.75e+16)
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * a));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * 4.0) * Float64(b * Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0))))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 1.75e+16)
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * a);
	else
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 1.75e+16], 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[(0.25 * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(b * N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.75e16

   1. Initial program 3.0%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 23.9%

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

   6. Taylor expanded in angle around 0

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

      1. *-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 20.0%

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

   8. Simplified 20.0%

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

   if 1.75e16 < y-scale

   1. Initial program 1.8%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 55.9%

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

   7. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right)\right)\right), \frac{1}{4}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      18. PI-lowering-PI.f64 61.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 61.6%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)}\right)\right) \cdot 0.25 \]

   11. Taylor expanded in a around 0

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

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

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

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

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

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

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

       4. unpow2 N/A

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

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

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

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

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

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

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

       8. unpow2 N/A

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

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

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

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

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

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

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

       12. associate-*r* N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

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

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

       18. PI-lowering-PI.f64 29.7%

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

   13. Simplified 29.7%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.75 \cdot 10^{+16}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot 4\right) \cdot \left(b \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 25.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.55 \cdot 10^{+20}:\\ \;\;\;\;0.25 \cdot \left(\left(a \cdot x-scale\_m\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \left(b \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 1.55e+20)
   (* 0.25 (* (* a x-scale_m) (* (sqrt 2.0) (sqrt 8.0))))
   (*
    0.25
    (*
     (* y-scale_m 4.0)
     (*
      b
      (+
       1.0
       (*
        (* angle angle)
        (+
         (* (* PI PI) -1.54320987654321e-5)
         (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0))))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.55e+20) {
		tmp = 0.25 * ((a * x_45_scale_m) * (sqrt(2.0) * sqrt(8.0)));
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0)))))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.55e+20) {
		tmp = 0.25 * ((a * x_45_scale_m) * (Math.sqrt(2.0) * Math.sqrt(8.0)));
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0)))))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 1.55e+20:
		tmp = 0.25 * ((a * x_45_scale_m) * (math.sqrt(2.0) * math.sqrt(8.0)))
	else:
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0)))))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 1.55e+20)
		tmp = Float64(0.25 * Float64(Float64(a * x_45_scale_m) * Float64(sqrt(2.0) * sqrt(8.0))));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * 4.0) * Float64(b * Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0))))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 1.55e+20)
		tmp = 0.25 * ((a * x_45_scale_m) * (sqrt(2.0) * sqrt(8.0)));
	else
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 1.55e+20], N[(0.25 * N[(N[(a * x$45$scale$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(b * N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.55e20

   1. Initial program 3.0%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 23.9%

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

   6. Taylor expanded in angle around 0

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

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

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 19.9%

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

   8. Simplified 19.9%

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

   if 1.55e20 < y-scale

   1. Initial program 1.8%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 55.9%

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

   7. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right)\right)\right), \frac{1}{4}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      18. PI-lowering-PI.f64 61.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 61.6%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)}\right)\right) \cdot 0.25 \]

   11. Taylor expanded in a around 0

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

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

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

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

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

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

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

       4. unpow2 N/A

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

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

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

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

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

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

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

       8. unpow2 N/A

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

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

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

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

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

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

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

       12. associate-*r* N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

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

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

       18. PI-lowering-PI.f64 29.7%

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

   13. Simplified 29.7%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.55 \cdot 10^{+20}:\\ \;\;\;\;0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot 4\right) \cdot \left(b \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.3% accurate, 20.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 3.05 \cdot 10^{+26}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5} + \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \left(b \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 3.05e+26)
   (*
    0.25
    (*
     (* y-scale_m 4.0)
     (+
      b
      (/
       (*
        0.5
        (*
         (* angle angle)
         (*
          (* PI PI)
          (+
           (* (* b b) -3.08641975308642e-5)
           (* (* a a) 3.08641975308642e-5)))))
       b))))
   (*
    0.25
    (*
     (* y-scale_m 4.0)
     (*
      b
      (+
       1.0
       (*
        (* angle angle)
        (+
         (* (* PI PI) -1.54320987654321e-5)
         (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0))))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 3.05e+26) {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)));
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0)))))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 3.05e+26) {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((Math.PI * Math.PI) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)));
	} else {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0)))))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 3.05e+26:
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((math.pi * math.pi) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)))
	else:
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0)))))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 3.05e+26)
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * 4.0) * Float64(b + Float64(Float64(0.5 * Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * Float64(Float64(Float64(b * b) * -3.08641975308642e-5) + Float64(Float64(a * a) * 3.08641975308642e-5))))) / b))));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * 4.0) * Float64(b * Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0))))))));
	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 <= 3.05e+26)
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((pi * pi) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)));
	else
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b * (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 3.05e+26], N[(0.25 * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(b + N[(N[(0.5 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(b * N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.0500000000000001e26

   1. Initial program 3.3%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 18.6%

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

   7. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

   10. Simplified 14.1%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \color{blue}{\left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(-3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\right)}{b}\right)}\right) \cdot 0.25 \]

   if 3.0500000000000001e26 < b

   1. Initial program 0.6%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 28.7%

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

   7. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right)\right)\right), \frac{1}{4}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      18. PI-lowering-PI.f64 25.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 25.4%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)}\right)\right) \cdot 0.25 \]

   11. Taylor expanded in a around 0

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

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

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

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

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

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

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

       4. unpow2 N/A

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

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

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

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

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

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

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

       8. unpow2 N/A

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

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

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

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

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

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

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

       12. associate-*r* N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

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

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

       18. PI-lowering-PI.f64 25.3%

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

   13. Simplified 25.3%

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

4. Final simplification 16.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.05 \cdot 10^{+26}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot 4\right) \cdot \left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5} + \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot 4\right) \cdot \left(b \cdot \left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.1% accurate, 21.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{+25}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5} + \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right) \cdot \left(y-scale\_m \cdot b\right)\\ \end{array} \end{array} \]

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 2.3e+25)
   (*
    0.25
    (*
     (* y-scale_m 4.0)
     (+
      b
      (/
       (*
        0.5
        (*
         (* angle angle)
         (*
          (* PI PI)
          (+
           (* (* b b) -3.08641975308642e-5)
           (* (* a a) 3.08641975308642e-5)))))
       b))))
   (*
    (+
     1.0
     (*
      (* angle angle)
      (+
       (* (* PI PI) -1.54320987654321e-5)
       (* (* (* angle angle) 3.969161205100849e-11) (pow PI 4.0)))))
    (* 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 <= 2.3e+25) {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)));
	} else {
		tmp = (1.0 + ((angle * angle) * (((((double) M_PI) * ((double) M_PI)) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * pow(((double) M_PI), 4.0))))) * (y_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 tmp;
	if (b <= 2.3e+25) {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((Math.PI * Math.PI) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)));
	} else {
		tmp = (1.0 + ((angle * angle) * (((Math.PI * Math.PI) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * Math.pow(Math.PI, 4.0))))) * (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 <= 2.3e+25:
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((math.pi * math.pi) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)))
	else:
		tmp = (1.0 + ((angle * angle) * (((math.pi * math.pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * math.pow(math.pi, 4.0))))) * (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 <= 2.3e+25)
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * 4.0) * Float64(b + Float64(Float64(0.5 * Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * Float64(Float64(Float64(b * b) * -3.08641975308642e-5) + Float64(Float64(a * a) * 3.08641975308642e-5))))) / b))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(angle * angle) * Float64(Float64(Float64(pi * pi) * -1.54320987654321e-5) + Float64(Float64(Float64(angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0))))) * Float64(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 <= 2.3e+25)
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((pi * pi) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)));
	else
		tmp = (1.0 + ((angle * angle) * (((pi * pi) * -1.54320987654321e-5) + (((angle * angle) * 3.969161205100849e-11) * (pi ^ 4.0))))) * (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, 2.3e+25], N[(0.25 * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(b + N[(N[(0.5 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.54320987654321e-5), $MachinePrecision] + N[(N[(N[(angle * angle), $MachinePrecision] * 3.969161205100849e-11), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale$95$m * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2999999999999998e25

   1. Initial program 3.3%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 18.6%

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

   7. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

   10. Simplified 14.1%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \color{blue}{\left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(-3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\right)}{b}\right)}\right) \cdot 0.25 \]

   if 2.2999999999999998e25 < b

   1. Initial program 0.6%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 28.7%

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

   7. Applied egg-rr 24.7%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)}\right)\right)\right), \frac{1}{4}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left({angle}^{2} \cdot \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({angle}^{2}\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left(\frac{-1}{64800} \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\frac{1}{25194240000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\left(\frac{1}{25194240000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

      18. PI-lowering-PI.f64 25.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), 180\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{64800}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{25194240000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), 4\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   10. Simplified 25.4%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \color{blue}{\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(3.969161205100849 \cdot 10^{-11} \cdot \left(angle \cdot angle\right)\right) \cdot {\pi}^{4}\right)\right)}\right)\right) \cdot 0.25 \]

   11. Taylor expanded in a around 0

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

       1. associate-*r* N/A

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

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

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

       3. *-commutative N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

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

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

       11. unpow2 N/A

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

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

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

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

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

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

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

       15. associate-*r* N/A

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

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

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

   13. Simplified 25.4%

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

4. Final simplification 16.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{+25}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot 4\right) \cdot \left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5} + \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.54320987654321 \cdot 10^{-5} + \left(\left(angle \cdot angle\right) \cdot 3.969161205100849 \cdot 10^{-11}\right) \cdot {\pi}^{4}\right)\right) \cdot \left(y-scale \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 25.5% accurate, 76.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5} + \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

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 2.7e+107)
   (*
    0.25
    (*
     (* y-scale_m 4.0)
     (+
      b
      (/
       (*
        0.5
        (*
         (* angle angle)
         (*
          (* PI PI)
          (+
           (* (* b b) -3.08641975308642e-5)
           (* (* a a) 3.08641975308642e-5)))))
       b))))
   (* 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 <= 2.7e+107) {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)));
	} else {
		tmp = y_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 tmp;
	if (b <= 2.7e+107) {
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((Math.PI * Math.PI) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)));
	} 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 <= 2.7e+107:
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((math.pi * math.pi) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)))
	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 <= 2.7e+107)
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * 4.0) * Float64(b + Float64(Float64(0.5 * Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * Float64(Float64(Float64(b * b) * -3.08641975308642e-5) + Float64(Float64(a * a) * 3.08641975308642e-5))))) / b))));
	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 <= 2.7e+107)
		tmp = 0.25 * ((y_45_scale_m * 4.0) * (b + ((0.5 * ((angle * angle) * ((pi * pi) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))))) / b)));
	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, 2.7e+107], N[(0.25 * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(b + N[(N[(0.5 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000001e107

   1. Initial program 3.1%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

   10. Simplified 13.9%

       \[\leadsto \left(\left(y-scale \cdot 4\right) \cdot \color{blue}{\left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(-3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\right)}{b}\right)}\right) \cdot 0.25 \]

   if 2.7000000000000001e107 < b

   1. Initial program 0.3%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 36.8%

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

   7. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 30.4%

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

   10. Simplified 30.4%

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

4. Final simplification 16.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+107}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot 4\right) \cdot \left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5} + \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 21.8% accurate, 81.1× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+107}:\\ \;\;\;\;y-scale\_m \cdot b + \frac{0.5 \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \left(\left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5} + \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(y-scale\_m \cdot \left(angle \cdot angle\right)\right)\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

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 1.2e+107)
   (+
    (* y-scale_m b)
    (/
     (*
      0.5
      (*
       (*
        (* PI PI)
        (+ (* (* b b) -3.08641975308642e-5) (* (* a a) 3.08641975308642e-5)))
       (* y-scale_m (* angle angle))))
     b))
   (* 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 <= 1.2e+107) {
		tmp = (y_45_scale_m * b) + ((0.5 * (((((double) M_PI) * ((double) M_PI)) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))) * (y_45_scale_m * (angle * angle)))) / b);
	} else {
		tmp = y_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 tmp;
	if (b <= 1.2e+107) {
		tmp = (y_45_scale_m * b) + ((0.5 * (((Math.PI * Math.PI) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))) * (y_45_scale_m * (angle * angle)))) / b);
	} 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 <= 1.2e+107:
		tmp = (y_45_scale_m * b) + ((0.5 * (((math.pi * math.pi) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))) * (y_45_scale_m * (angle * angle)))) / b)
	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 <= 1.2e+107)
		tmp = Float64(Float64(y_45_scale_m * b) + Float64(Float64(0.5 * Float64(Float64(Float64(pi * pi) * Float64(Float64(Float64(b * b) * -3.08641975308642e-5) + Float64(Float64(a * a) * 3.08641975308642e-5))) * Float64(y_45_scale_m * Float64(angle * angle)))) / b));
	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 <= 1.2e+107)
		tmp = (y_45_scale_m * b) + ((0.5 * (((pi * pi) * (((b * b) * -3.08641975308642e-5) + ((a * a) * 3.08641975308642e-5))) * (y_45_scale_m * (angle * angle)))) / b);
	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, 1.2e+107], N[(N[(y$45$scale$95$m * b), $MachinePrecision] + N[(N[(0.5 * N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale$95$m * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2e107

   1. Initial program 3.1%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 18.1%

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

   7. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{angle}^{2} \cdot \left(y-scale \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{b} + b \cdot y-scale} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot y-scale + \color{blue}{\frac{1}{2} \cdot \frac{{angle}^{2} \cdot \left(y-scale \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{b}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot y-scale\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{{angle}^{2} \cdot \left(y-scale \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{b}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(y-scale \cdot b\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{{angle}^{2} \cdot \left(y-scale \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{b}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y-scale, b\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{{angle}^{2} \cdot \left(y-scale \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{b}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y-scale, b\right), \left(\frac{\frac{1}{2} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right)}{\color{blue}{b}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y-scale, b\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({angle}^{2} \cdot \left(y-scale \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right)\right), \color{blue}{b}\right)\right) \]

   10. Simplified 12.9%

       \[\leadsto \color{blue}{y-scale \cdot b + \frac{0.5 \cdot \left(\left(\left(angle \cdot angle\right) \cdot y-scale\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(-3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\right)}{b}} \]

   if 1.2e107 < b

   1. Initial program 0.3%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 36.8%

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

   7. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 30.4%

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

   10. Simplified 30.4%

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

4. Final simplification 15.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+107}:\\ \;\;\;\;y-scale \cdot b + \frac{0.5 \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \left(\left(b \cdot b\right) \cdot -3.08641975308642 \cdot 10^{-5} + \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(y-scale \cdot \left(angle \cdot angle\right)\right)\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 19.3% accurate, 91.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq -30000000:\\ \;\;\;\;\left(b \cdot 0.25\right) \cdot \left(x-scale\_m \cdot \left(\left(\left(angle \cdot angle\right) \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \frac{\pi}{180}\right) \cdot \left(angle \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.25 \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= angle -30000000.0)
   (*
    (* b 0.25)
    (*
     x-scale_m
     (*
      (+
       (* (* angle angle) (* -2.8577960676726107e-8 (* PI (* PI PI))))
       (/ PI 180.0))
      (* angle 4.0))))
   (* b (* 0.25 (* y-scale_m 4.0)))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (angle <= -30000000.0) {
		tmp = (b * 0.25) * (x_45_scale_m * ((((angle * angle) * (-2.8577960676726107e-8 * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))) + (((double) M_PI) / 180.0)) * (angle * 4.0)));
	} else {
		tmp = b * (0.25 * (y_45_scale_m * 4.0));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (angle <= -30000000.0) {
		tmp = (b * 0.25) * (x_45_scale_m * ((((angle * angle) * (-2.8577960676726107e-8 * (Math.PI * (Math.PI * Math.PI)))) + (Math.PI / 180.0)) * (angle * 4.0)));
	} else {
		tmp = b * (0.25 * (y_45_scale_m * 4.0));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if angle <= -30000000.0:
		tmp = (b * 0.25) * (x_45_scale_m * ((((angle * angle) * (-2.8577960676726107e-8 * (math.pi * (math.pi * math.pi)))) + (math.pi / 180.0)) * (angle * 4.0)))
	else:
		tmp = b * (0.25 * (y_45_scale_m * 4.0))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (angle <= -30000000.0)
		tmp = Float64(Float64(b * 0.25) * Float64(x_45_scale_m * Float64(Float64(Float64(Float64(angle * angle) * Float64(-2.8577960676726107e-8 * Float64(pi * Float64(pi * pi)))) + Float64(pi / 180.0)) * Float64(angle * 4.0))));
	else
		tmp = Float64(b * Float64(0.25 * Float64(y_45_scale_m * 4.0)));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (angle <= -30000000.0)
		tmp = (b * 0.25) * (x_45_scale_m * ((((angle * angle) * (-2.8577960676726107e-8 * (pi * (pi * pi)))) + (pi / 180.0)) * (angle * 4.0)));
	else
		tmp = b * (0.25 * (y_45_scale_m * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[angle, -30000000.0], N[(N[(b * 0.25), $MachinePrecision] * N[(x$45$scale$95$m * N[(N[(N[(N[(angle * angle), $MachinePrecision] * N[(-2.8577960676726107e-8 * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision] * N[(angle * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(0.25 * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < -3e7

   1. Initial program 0.1%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 20.2%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 9.1%

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

   8. Simplified 9.1%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\left(\frac{-1}{34992000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{34992000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       17. PI-lowering-PI.f64 18.8%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   11. Simplified 18.8%

       \[\leadsto 0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(\left(\color{blue}{\left(angle \cdot \left(\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right) + 0.005555555555555556 \cdot \pi\right)\right)} \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \color{blue}{\left(x-scale \cdot \left(\left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{4} \cdot b\right) \cdot \color{blue}{\left(x-scale \cdot \left(\left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot b\right), \color{blue}{\left(x-scale \cdot \left(\left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\color{blue}{x-scale} \cdot \left(\left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(x-scale, \color{blue}{\left(\left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(x-scale, \left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)}\right)\right)\right) \]

       7. sqrt-unprod N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(x-scale, \left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2 \cdot 8}\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(x-scale, \left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{16}\right)\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(x-scale, \left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 4\right)\right)\right) \]

   13. Applied egg-rr 18.8%

       \[\leadsto \color{blue}{\left(0.25 \cdot b\right) \cdot \left(x-scale \cdot \left(\left(\left(angle \cdot angle\right) \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \frac{\pi}{180}\right) \cdot \left(angle \cdot 4\right)\right)\right)} \]

   if -3e7 < angle

   1. Initial program 3.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 18.0%

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

   5. Simplified 18.0%

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

      1. *-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-*r* N/A

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

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

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

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

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

      5. associate-*l* N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{16}\right), \frac{1}{4}\right), b\right) \]

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 18.1%

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

   7. Applied egg-rr 18.1%

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

4. Final simplification 18.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -30000000:\\ \;\;\;\;\left(b \cdot 0.25\right) \cdot \left(x-scale \cdot \left(\left(\left(angle \cdot angle\right) \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \frac{\pi}{180}\right) \cdot \left(angle \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.25 \cdot \left(y-scale \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 19.3% accurate, 91.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq -235000000:\\ \;\;\;\;0.25 \cdot \left(x-scale\_m \cdot \left(b \cdot \left(\left(\left(angle \cdot angle\right) \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \frac{\pi}{180}\right) \cdot \left(angle \cdot 4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.25 \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= angle -235000000.0)
   (*
    0.25
    (*
     x-scale_m
     (*
      b
      (*
       (+
        (* (* angle angle) (* -2.8577960676726107e-8 (* PI (* PI PI))))
        (/ PI 180.0))
       (* angle 4.0)))))
   (* b (* 0.25 (* y-scale_m 4.0)))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (angle <= -235000000.0) {
		tmp = 0.25 * (x_45_scale_m * (b * ((((angle * angle) * (-2.8577960676726107e-8 * (((double) M_PI) * (((double) M_PI) * ((double) M_PI))))) + (((double) M_PI) / 180.0)) * (angle * 4.0))));
	} else {
		tmp = b * (0.25 * (y_45_scale_m * 4.0));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (angle <= -235000000.0) {
		tmp = 0.25 * (x_45_scale_m * (b * ((((angle * angle) * (-2.8577960676726107e-8 * (Math.PI * (Math.PI * Math.PI)))) + (Math.PI / 180.0)) * (angle * 4.0))));
	} else {
		tmp = b * (0.25 * (y_45_scale_m * 4.0));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if angle <= -235000000.0:
		tmp = 0.25 * (x_45_scale_m * (b * ((((angle * angle) * (-2.8577960676726107e-8 * (math.pi * (math.pi * math.pi)))) + (math.pi / 180.0)) * (angle * 4.0))))
	else:
		tmp = b * (0.25 * (y_45_scale_m * 4.0))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (angle <= -235000000.0)
		tmp = Float64(0.25 * Float64(x_45_scale_m * Float64(b * Float64(Float64(Float64(Float64(angle * angle) * Float64(-2.8577960676726107e-8 * Float64(pi * Float64(pi * pi)))) + Float64(pi / 180.0)) * Float64(angle * 4.0)))));
	else
		tmp = Float64(b * Float64(0.25 * Float64(y_45_scale_m * 4.0)));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (angle <= -235000000.0)
		tmp = 0.25 * (x_45_scale_m * (b * ((((angle * angle) * (-2.8577960676726107e-8 * (pi * (pi * pi)))) + (pi / 180.0)) * (angle * 4.0))));
	else
		tmp = b * (0.25 * (y_45_scale_m * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[angle, -235000000.0], N[(0.25 * N[(x$45$scale$95$m * N[(b * N[(N[(N[(N[(angle * angle), $MachinePrecision] * N[(-2.8577960676726107e-8 * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision] * N[(angle * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(0.25 * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < -2.35e8

   1. Initial program 0.1%

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

   3. Taylor expanded in y-scale around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. distribute-lft-out N/A

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

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

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

   5. Simplified 20.2%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 9.1%

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

   8. Simplified 9.1%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\left(\frac{-1}{34992000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{34992000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

       17. PI-lowering-PI.f64 18.8%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{180}, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   11. Simplified 18.8%

       \[\leadsto 0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(\left(\color{blue}{\left(angle \cdot \left(\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right) + 0.005555555555555556 \cdot \pi\right)\right)} \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \color{blue}{\left(b \cdot x-scale\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\left(\left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot b\right) \cdot \color{blue}{x-scale}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\left(\left(\left(angle \cdot \left(\left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot b\right), \color{blue}{x-scale}\right)\right) \]

   13. Applied egg-rr 18.8%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(\left(\left(angle \cdot angle\right) \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \frac{\pi}{180}\right) \cdot \left(angle \cdot 4\right)\right) \cdot b\right) \cdot x-scale\right)} \]

   if -2.35e8 < angle

   1. Initial program 3.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 18.0%

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

   5. Simplified 18.0%

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

      1. *-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-*r* N/A

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

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

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

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

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

      5. associate-*l* N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{16}\right), \frac{1}{4}\right), b\right) \]

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 18.1%

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

   7. Applied egg-rr 18.1%

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

4. Final simplification 18.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -235000000:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(b \cdot \left(\left(\left(angle \cdot angle\right) \cdot \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \frac{\pi}{180}\right) \cdot \left(angle \cdot 4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.25 \cdot \left(y-scale \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 17.9% accurate, 393.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ b \cdot \left(0.25 \cdot \left(y-scale\_m \cdot 4\right)\right) \end{array} \]

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
 (* b (* 0.25 (* y-scale_m 4.0))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	return b * (0.25 * (y_45_scale_m * 4.0));
}

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 = b * (0.25d0 * (y_45scale_m * 4.0d0))
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 b * (0.25 * (y_45_scale_m * 4.0));
}

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 b * (0.25 * (y_45_scale_m * 4.0))

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(b * Float64(0.25 * Float64(y_45_scale_m * 4.0)))
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 = b * (0.25 * (y_45_scale_m * 4.0));
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[(b * N[(0.25 * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.7%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r* N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

   8. sqrt-lowering-sqrt.f64 17.5%

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

5. Simplified 17.5%

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

   1. *-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-*r* N/A

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

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

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

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

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

   5. associate-*l* N/A

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

   6. sqrt-unprod N/A

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{16}\right), \frac{1}{4}\right), b\right) \]

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 17.6%

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

7. Applied egg-rr 17.6%

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

8. Final simplification 17.6%

   \[\leadsto b \cdot \left(0.25 \cdot \left(y-scale \cdot 4\right)\right) \]
9. Add Preprocessing

Alternative 13: 18.0% 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 2.7%

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

3. Taylor expanded in x-scale around 0

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   7. distribute-lft-out N/A

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

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

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

5. Simplified 20.8%

   \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right) + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)\right)}} \]

7. Applied egg-rr 23.7%

   \[\leadsto \color{blue}{\left(\left(y-scale \cdot 4\right) \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right), b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)\right) \cdot 0.25} \]

8. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{b \cdot y-scale} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto y-scale \cdot \color{blue}{b} \]

   2. *-lowering-*.f64 17.6%

      \[\leadsto \mathsf{*.f64}\left(y-scale, \color{blue}{b}\right) \]

10. Simplified 17.6%

    \[\leadsto \color{blue}{y-scale \cdot b} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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))))